Chapter 5

Geometric Applications of Differential Calculus

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Section 5.1Equations of Curves and Their Types[41/68%]

Problem 1874

A flat eccentric disk is mounted on a horizontal axis. What shape must it have so that a vertical rod resting against it performs harmonic oscillatory motion when the axis rotates uniformly?

Problem 1875

The same question, but the vertical rod must perform uniform motion up and down. (This is how the tube of certain microscopes moves.)

Problem 1876

An obliquely cut cylinder is covered with coloring substance and rolls along a plane. What curve bounds the colored part of the plane?

Problem 1877

Prove that the curves $r = e^{a\varphi }$ and $r = ce^{a\varphi }$ are geometrically similar — one is transformed into the other by enlarging the scale of the drawing. At the same time they are congruent, transforming into each other under a rotation through a certain angle.

Problem 1878

Prove that the figures enclosed between the $x$ -axis and the curves $y = e^{x}$ and $y = ae^{x}$ can be made to coincide by translating one of these curves along the $x$ -axis.

Problem 1879

A straight line of given length $l$ slides with its ends $A$ and $B$ along the axes $Ox$ and $Oy$ . From the vertex $C$ of the rectangle $OACB$ a perpendicular is dropped onto the line. Find the locus of the foot of this perpendicular.

Problem 1880

The same question for the endpoint of the perpendicular dropped from the origin. (The four-petaled rose.)

Problem 1881

A segment $OO_{1} = a$ is given. From an arbitrary ray through the point $O$ , a perpendicular $O_{1}M_{1}$ is dropped from the point $O_{1}$ , and from the foot of this perpendicular $M_{1}$ a perpendicular $M_{1}M$ is dropped onto the ray symmetric to the first one with respect to the line $OO_{1}$ . Find the locus of the feet $M$ of the second perpendicular.

Problem 1882

From a point $M(\xi , \eta )$ on the circle $x^{2}+y^{2}=a^{2}$ , a line $y=\eta$ is drawn to its intersection with the $y$ -axis at a point $N$ . From the point $(a, 0)$ a line is drawn to the point $N$ . Find the locus of the point of its intersection with the ordinate of the point $M$ .

Problem 1883

A right angle rotates about its vertex $A(a, 0)$ . From the origin, through the point where one side of the angle meets the circle $x^{2}+y^{2}=r^{2}$ , a line is drawn to its intersection with the other side of the angle. Find the locus of the point of intersection.

Problem 1884

A right triangle has legs $AC = a$ and $BC = 2a$ . The point $B$ slides along the axis $Ox$ , and the point $A$ slides along the circle $x^{2}+y^{2}=a^{2}$ . Find the locus of the vertices of the right angle.

Problem 1885

Find the locus of the points of tangency of lines from the origin that are tangent to circles of radius $a$ with centers on the axis $Ox$ .

Problem 1886

Two equal parabolas rotate about their vertices so that the four points of their intersection always lie on a circle. Find the locus of the center of the circle.

Problem 1887

Find the locus of the vertex of an equilateral (rectangular) hyperbola with a given center, passing through a given point.

Problem 1888

The same question for the focus of the hyperbola.

Problem 1889

A hyperbola with semi-axis $a$ passes through two given points at distance $2c$ apart. Find the locus of the center of the hyperbola, if it is equilateral.

Problem 1890

Find the locus of the midpoints of the segments of normals to the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ lying inside the ellipse.

Problem 1891

Two rods of length $\sqrt{2}$ rotate about their fixed endpoints at the points $(1,0)$ and $(-1,0)$ . Their other endpoints are joined by a hinge to a rod of length $2$ . Determine what curve the midpoint of the connecting rod describes.

Problem 1892

The Peaucellier–Lipkin inversor consists of a rhombus $ABCD$ with side equal to $a$ . The points $A$ and $C$ are joined to a fixed point $O$ by rods $OA$ and $OC$ of length $b$ , where $b > a$ .

The sides of the rhombus and the segments $OA$ and $OC$ are rods joined by hinges. Prove that the points $O$ , $B$ , and $D$ lie on one line, and moreover $OB \cdot OD = b^{2}-a^{2}$ , so that the motion of a vertex of the rhombus along a given curve allows one to carry out an inversion.

Problem 1893

Prove that when the point $B$ in the Peaucellier–Lipkin inversor moves along a circle passing through the point $O$ , the point $D$ describes a straight line.

Problem 1894

What curve is represented by the parametric equations $x = t^{2}-t+1$ , $y = t^{2}+t+1$ ?

Problem 1895

What curve is represented by the parametric equations $x = t^{2}-2t+3$ , $y = t^{2}-2t+1$ ?

Problem 1896

What curve is represented by the parametric equations $x = a\sin^{2}t$ , $y = b\cos^{2}t$ ?

Problem 1897

What curve is represented by the parametric equations $x = a\sin^{4}t$ , $y = a\cos^{4}t$ ?

Problem 1898

On the curve $x = t^{2}-2t+3$ , $y = t^{2}+2t-1$ , find the leftmost and the lowest points.

Problem 1899

Using the identity $\cos^{2}t+\sin^{2}t=1$ , represent the equation of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ in parametric form.

Problem 1900

Using the identity $e^{-t}\cdot e^{t}=1$ , represent the equation of the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}=1$ in parametric form.

Problem 1901

Represent the equation of the circle $x^{2}+y^{2}=a^{2}$ in parametric form, taking as parameter the slope of a chord of the circle passing through the point $(-a, 0)$ .

Problem 1902

Prove that the coordinates of the points of a second-order curve can be expressed rationally in terms of the slope of a variable chord of the curve passing through some chosen point of the curve.

Problem 1903

A curve $F(x,y)=0$ whose points' coordinates can be expressed rationally in terms of a parameter is called unicursal. Prove that the curves $r = f(\cos \varphi , \sin \varphi )$ , where $f$ is a rational function, are unicursal.

Problem 1904

Prove that the cissoid $y^{2} = \dfrac {x^{3}}{2a-x}$ is unicursal, by expressing the coordinates of its points in terms of the slope of the chord joining the origin to a given point of the curve.

Problem 1905

Solving simultaneously the equation of the folium of Descartes $x^{3}+y^{3}=3axy$ and the equation of the auxiliary line $y = tx$ , obtain the parametric equations of the folium of Descartes:

x = \frac{3at}{1+t^{3}}, \qquad y = \frac{3at^{2}}{1+t^{3}}.

Problem 1906

Prove that three distinct points of the folium of Descartes, corresponding to the values $t_{1}$ , $t_{2}$ , and $t_{3}$ of the parameter, lie on one line if and only if $t_{1}t_{2}t_{3} = -1$ .

Problem 1907

Prove that the coordinates of the points of the cardioid $r = a(1+\cos \varphi )$ can be expressed in terms of some parameter $t$ by the formulas:

x = \frac{2a(1-t^{2})}{(1+t^{2})^{2}}, \qquad y = \frac{4at}{(1+t^{2})^{2}}.

Problem 1908

Under what condition do three points of the cardioid, corresponding to the values $t_{1}$ , $t_{2}$ , $t_{3}$ of the parameter (see the previous problem), lie on one line?

Problem 1909

Prove that the coordinates of the points of the strophoid $(2a-x)y^{2} = x(x-a)^{2}$ can be expressed rationally in terms of some parameter $t$ by the formulas:

x = \frac{2at^{2}}{1+t^{2}}, \qquad y = \frac{at(t^{2}-1)}{1+t^{2}}.

Problem 1910

Under what condition do the points of the strophoid corresponding to the values $t_{1}$ , $t_{2}$ , $t_{3}$ of the parameter (see the previous problem) lie on one line?

Problem 1911

For what values of $t_{1}$ , $t_{2}$ , $t_{3}$ , $t_{4}$ do the corresponding points of the strophoid (see problem 1909) lie on one circle?

Problem 1912

Prove that the four points of the cissoid

x = \frac{a}{t^{2}+1}, \qquad y = \frac{a}{t(t^{2}+1)},

corresponding to the values $t_{1}$ , $t_{2}$ , $t_{3}$ , and $t_{4}$ of the parameter, lie on one circle provided that $t_{1}+t_{2}+t_{3}+t_{4}=0$ .

Problem 1913

Express rationally, in terms of an auxiliary parameter, the coordinates of the points of the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ .

Problem 1914

Show that the curve $x^{y}=y^{x}$ splits into the bisector of the first coordinate angle and the curve

x = \left(1+\frac{1}{n}\right)^{n}, \qquad y = \left(1+\frac{1}{n}\right)^{n+1},

where $n$ is a variable parameter.

Section 5.2The Tangent and the Normal[90/54%]

Problem 1915

Show that the curves

y = a\sin \frac{x}{a}, \qquad y = a\tg \frac{x}{a}, \qquad y = a\ln \frac{x}{a}

intersect the axis $Ox$ at equal angles, independent of the value of $a$ .

Problem 1916

Find the angle at which the curve

y = \frac{x+a_{2}x^{2}+a_{3}x^{3}+\ldots +a_{n}x^{n}}{1+b_{1}x+b_{2}x^{2}+\ldots +b_{m}x^{m}}

intersects the axis $Oy$ .

Problem 1917

For what value of $a$ does the curve $y = \dfrac {x^{3}+ax}{b}$ intersect the axis $Oy$ at an angle $\dfrac {\pi }{4}$ ?

Problem 1918

The same question for the curve $y = \dfrac {ax}{1+bx^{2}}$ .

Problem 1919

Which of the lines through the origin intersects the hyperbola $xy = a^{2}$ at a right angle?

Problem 1920

The same question for the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ .

Problem 1921

Find the equation of a sinusoid intersecting the axis $Ox$ at the origin at an angle $\dfrac {\pi }{4}$ , and at the point $(a,0)$ at an angle $\dfrac {3\pi }{4}$ .

Problem 1922

Prove that the angle between the radius vector drawn from the origin to a point $(r,\varphi )$ of the curve $r^{n} = a^{n}\cos n\varphi$ and the tangent to this curve at that point equals $n\varphi +\dfrac {\pi }{2}$ .

Problem 1923

Prove that the curve $r = e^{a\varphi }$ intersects all radius vectors issuing from the origin at the same angle.

Problem 1924

Prove that the angle between the tangent to the Archimedean spiral $r = a\varphi$ and the radius vector drawn to the point of tangency from the origin tends to $\dfrac {\pi }{2}$ as $\varphi \rightarrow \infty$ .

Problem 1925

Find the angle between the tangent to the cardioid $r = a(1+\cos \varphi )$ and the radius vector of the point of tangency.

Problem 1926

Find the equation of the tangent to the curve $y = x\ln x+1$ at the point with ordinate $1$ .

Problem 1927

Find the equation of the tangent at the point $(1,-2)$ to the curve

2x^{3}-x^{2}y^{2}-3x+y+7=0 .

Problem 1928

Draw the normal to the curve $x^{3}+y^{2}+2x-6=0$ at the point where $y=3$ .

Problem 1929

Find the normal to the witch of Agnesi $x^{2}y = a^{2}(a-y)$ that is parallel to the line $y=2x$ .

Problem 1930

Find the normal to the curve $y = a\ln \cos \dfrac {x}{a}$ at the point $x=2\pi a$ .

Problem 1931

Find the normal to the cardioid $r = a(1+\cos \varphi )$ that makes an angle $\dfrac {\pi }{4}$ with the polar axis.

Problem 1932

On the curve $xy^{2}=2a^{3}$ , find the points at which the normal passes through the origin.

Problem 1933

For the curve $x^{3}-y^{3}=3x^{2}$ , find the tangent that is parallel to the line $y=x$ .

Problem 1934

Compute the distance from the origin to the tangent to the curve $x^{2}y=a^{3}$ drawn at the point with abscissa $x_{0}$ .

Problem 1935

Find the tangent to the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ that is farthest from the origin.

Problem 1936

Prove that the tangents to the curves $y = af(x)$ at points having a common abscissa of the point of tangency meet at one point.

Problem 1937

Prove that at points with equal abscissas, the subnormals to the curves $y=f(x)$ and $y=\sqrt{f^{2}(x)+a^{2}}$ are equal.

Problem 1938

Prove that for the curves $y = ax^{n}$ the ratio of the subtangent to the abscissa of the point of tangency equals $\dfrac {1}{n}$ .

Problem 1939

Prove that for the curve $y = a\ln (x^{2}-a^{2})$ the sum of the lengths of the tangent and the subtangent is proportional to the quantity $xy$ .

Problem 1940

Prove that the length of the tangent to the tractrix

\frac{x+\sqrt{a^{2}-y^{2}}}{a} = \ln \frac{a+\sqrt{a^{2}-y^{2}}}{y}

equals $a$ .

Problem 1941

Prove that all normals to the curve

x = a(\cos t+t\sin t), \qquad y = a(\sin t-t\cos t)

are equidistant from the origin.

Problem 1942

Prove that the segment of the normal to the curve

x = 2a\sin t+a\sin t\cos ^{2}t, \qquad y = -a\cos ^{3}t,

lying between the coordinate axes, equals $2a$ .

Problem 1943

Prove that for the curve $x = 2a\ln \sin t-2a\sin^{2}t$ , $y = a\sin 2t$ the segment of the axis $Ox$ between the tangent and the normal equals $2a$ .

Problem 1944

Prove that the circle $x^{2}+y^{2}=a^{2}$ and the normals to the epicycloid

x = a\left((1+\lambda )\cos t-\lambda \cos \left(1+\frac{1}{\lambda }\right)t\right),

y = a\left((1+\lambda )\sin t-\lambda \sin \left(1+\frac{1}{\lambda }\right)t\right)

meet at the point $(a\cos t, a\sin t)$ .

Problem 1945

The same for the hypocycloid

x = a\left((1-\lambda )\cos t+\lambda \cos \left(1-\frac{1}{\lambda }\right)t\right),

y = a\left((1-\lambda )\sin t+\lambda \sin \left(1-\frac{1}{\lambda }\right)t\right).

Problem 1946

Determine the length of the segment of the tangent to the astroid

x = a\cos ^{3}t, \qquad y = a\sin ^{3}t

between the coordinate axes.

Problem 1947

Find the segment of the normal to the curve

x = (a+b)\sin t-a\sin ^{3}t, \qquad y = (a-b)\cos t+a\sin ^{2}t\cos t,

lying between the coordinate axes.

Problem 1948

Find the distance from the origin to the tangent to the curve $x = a(\cos t+t\sin t)$ , $y = a(\sin t-t\cos t)$ .

Problem 1949

Compute the distance from the origin to the tangent to the curve $r = ae^{k\varphi }$ .

Problem 1950

Compute the product of the distances from the foci of the ellipse $x = a\cos t$ , $y = b\sin t$ to the tangent to it.

Problem 1951

Compute the length of the segment of the normal to the ellipse $x = a\cos t$ , $y = b\sin t$ lying inside this ellipse.

Problem 1952

The normal drawn at a point $M$ of the curve $xy=a^{2}$ meets the line $y=x$ at a point $N$ . Find the ratio $OM : MN$ .

Problem 1953

Prove that the area of the trapezoid formed by the tangent to the curve $3axy = x^{3}+2a^{3}$ , the ordinate of the point of tangency, and the coordinate axes does not depend on the choice of the point of tangency.

Problem 1954

Prove that the tangent to the curve

x = a(1-\cos t), \qquad y = a(1-\sin t)

forms, together with the coordinate axes, a triangle whose perimeter is constant.

Problem 1955

Prove that the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ cuts off segments on the coordinate axes whose sum is constant.

Problem 1956

Prove that the angles between the tangents to the curves $r = f(\varphi )$ and $r = \dfrac {a}{f(\varphi )}$ and the radius vectors, at the same value of $\varphi$ , at the points of tangency, sum to $\pi$ .

Problem 1957

Prove that the tangents to the curves $y = \dfrac {\lambda \varphi (x)+\mu \varphi_{1}(x)}{\lambda +\mu }$ at points with equal abscissas of the point of tangency meet at one point, whose position does not depend on $\lambda$ and $\mu$ .

Problem 1958

Every radius vector of the curve $r = a\sin^{3}\dfrac {\varphi }{3}$ meets it at three points other than the pole. Prove that the tangents at these points form an equilateral triangle.

Problem 1959

At three points $M_{1}$ , $M_{2}$ , $M_{3}$ , lying on one line, on the folium of Descartes $x^{3}+y^{3}=3xy$ , tangents are drawn, meeting this curve again at points $P_{1}$ , $P_{2}$ , and $P_{3}$ . Prove that these points lie on one line.

Problem 1960

From a point $M$ on the strophoid $(2a-x)y^{2}=x(x-a)^{2}$ , lines are drawn tangent to the strophoid at points $P$ and $Q$ . Prove that the points $M$ , $P$ , and $Q$ lie on one circle passing through the origin. (See problems 1909, 1910, 1911.)

Problem 1961

For the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ , two pairs of tangents can be drawn parallel to a given direction. Prove that the lines joining the points of tangency of each pair make an angle $\dfrac {2\pi }{3}$ with each other.

Problem 1962

Find the length of the polar subtangent and subnormal for the spiral $r^{2}=a^{2}\varphi$ .

Problem 1963

Prove that the polar subtangents of the curves $r = f(\varphi )$ and $r = \dfrac {af(\varphi )}{a+f(\varphi )}$ are equal at points lying on the same radius vector.

Problem 1964

Prove that for the curves $r = f(\varphi )$ and $r = a+f(\varphi )$ , at points lying on the same radius vector, the polar subnormals are equal.

Problem 1965

Find the circle of radius $a$ tangent to the folium of Descartes $x^{3}+y^{3}-3axy=0$ at the point $\left(\dfrac {4a}{3}, \dfrac {2a}{3}\right)$ .

Problem 1966

Draw a circle of radius $3a$ normal to the cissoid $x(x^{2}+y^{2})-2ay^{2}=0$ at the point where $x=a$ .

Problem 1967

Find the circle passing through the pole and tangent to the Archimedean spiral $r=a\varphi$ at $\varphi =\dfrac {\pi }{4}$ .

Problem 1968

Show that the following pair of curves intersects at a right angle: $x^{2}-y^{2}=a^{2}$ , $xy=b^{2}$ .

Problem 1969

Show that the following pair of curves intersects at a right angle: $y^{2}=2ux+u^{2}$ , $y^{2}=-2vx+v^{2}$ .

Problem 1970

Show that the following pair of curves intersects at a right angle: $\dfrac {x^{2}}{a^{2}+\lambda }+\dfrac {y^{2}}{b^{2}+\lambda }=1$ , $\dfrac {x^{2}}{a^{2}-\mu }+\dfrac {y^{2}}{b^{2}-\mu }=1$ ; $a>\mu >b$ .

Problem 1971

Show that the following pair of curves intersects at a right angle: $(2a-x)y^{2}=x^{3}$ , $(x^{2}+y^{2})^{2}=b^{2}(2x^{2}+y^{2})$ .

Problem 1972

Show that the following pair of curves intersects at a right angle: $b^{2}x^{2}+a^{2}y^{2}=c$ , $y^{b}=c_{1}x^{a^{2}}$ .

Problem 1973

Show that the following pair of curves intersects at a right angle: $\sin y = ae^{-x}$ , $\cos y = be^{-x}$ .

Problem 1974

Show that the following pair of curves intersects at a right angle: $r^{n}=a^{n}\cos n\varphi$ , $r^{n}=b^{n}\sin n\varphi$ .

Problem 1975

Show that the following pair of curves intersects at a right angle: $r^{2}=\ln \tg \varphi$ , $r^{2}\cos 2\varphi +1=0$ .

Problem 1976

Prove that the curves

r^{2}\cos 2\varphi = a^{2}, \qquad r^{2}\cos (2\varphi +\alpha ) = b^{2}

intersect at an angle $\alpha$ .

Problem 1977

Prove that the curve

x = \left(1+\frac{1}{u}\right)^{u}, \qquad y = \left(1+\frac{1}{u}\right)^{u+1}

intersects the line $y=x$ at a right angle.

Problem 1978

Prove that the circles

x^{2}+y^{2}+ax+by+c=0, \qquad x^{2}+y^{2}+a_{1}x+b_{1}y+c_{1}=0

intersect at a right angle provided the condition $aa_{1}+bb_{1}=2(c+c_{1})$ holds.

Problem 1979

The geographic latitude is the angle between the normal to the Earth's surface at a given place and the plane of the equator. The geocentric latitude is the angle between that same plane and the radius vector from the Earth's center to the given point. Determine the greatest difference between these latitudes, taking the Earth's meridian to be an ellipse with $a=6400$ km, $a-b=21$ km.

Problem 1980

A curve $f(x,y)=0$ is called algebraic if $f(x,y)$ is a polynomial in $x$ and $y$ . By introducing powers of an auxiliary variable $z$ , this polynomial can be made homogeneous. Let $f(x,y,z)=0$ be the resulting equation. Prove that the equation of the tangent to the curve can be written in the form

\frac{\partial f}{\partial x}X+\frac{\partial f}{\partial y}Y+\frac{\partial f}{\partial z}Z=0,

where $(x,y)$ is the point of tangency, $(X,Y)$ is a point on the tangent, and the quantities $z$ and $Z$ , after the derivatives are found, are replaced by unity.

Problem 1981

The pedal curve of a curve with respect to a given point is the locus of the feet of perpendiculars dropped from the given point onto the tangents to the given curve. Find the pedal curve of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ with respect to the center.

Problem 1982

Find the pedal curve of the parabola $y^{2}=2px$ with respect to the vertex.

Problem 1983

Find the pedal curve of the parabola $y^{2}=2px$ with respect to the focus.

Problem 1984

Find the pedal curve of the hyperbola $x^{2}-y^{2}=a^{2}$ with respect to the origin.

Problem 1985

Find the pedal curve of the curve $\left(\dfrac {x}{a}\right)^{n}\pm \left(\dfrac {y}{b}\right)^{n}=1$ with respect to the point $(0,0)$ .

Problem 1986

Find the pedal curve of the curve $r=e^{a\varphi }$ with respect to the pole.

Problem 1987

Find the pedal curve of the curve $x^{2}+y^{2}=2ax$ with respect to the point $(0,0)$ (give the answer in polar coordinates).

Problem 1988

Find the pedal curve of the curve $r^{n}=a^{n}\cos n\varphi$ with respect to the pole.

Problem 1989

Find the pedal curve of the involute of a circle:

x = a(\cos t+t\sin t), \qquad y = a(\sin t-t\cos t)

with respect to the origin.

Problem 1990

Prove that the points of tangency of the tangents drawn from a given point to the logarithmic spiral $r=ae^{m\varphi }$ lie on a circle passing through the pole and the given point.

Problem 1991

From the origin a ray is drawn, meeting the strophoid $y^{2}(2a-x)=x(x-a)^{2}$ at points $M$ and $M_{1}$ . Find the locus of the points of intersection of the tangents to the strophoid at the points $M$ and $M_{1}$ .

Problem 1992

From a given point $M$ , three tangents can be drawn to the cardioid $r=a(1+\cos \varphi )$ . What is the locus of points $M$ for which the three points of tangency lie on one line?

Problem 1993

What is the locus of the endpoints of the polar subtangent for the hyperbolic spiral $r\varphi =a$ ?

Problem 1994

Find the locus of the vertices of a right angle whose sides are tangent to the hyperbola $b^{2}x^{2}-a^{2}y^{2}=a^{2}b^{2}$ .

Problem 1995

Find the locus of the vertices of a right angle whose sides are tangent to the astroid $x=a\cos^{3}t$ , $y=a\sin^{3}t$ .

Problem 1996

Find the locus of the vertices of a right angle whose sides are tangent to Neile's parabola $4y^{3}=27ax^{2}$ .

Problem 1997

Find the locus of the vertices of a right angle whose sides are tangent to the cycloid $x=a(t-\sin t)$ , $y=a(1-\cos t)$ .

Problem 1998

What is the locus of the vertices of an angle of given magnitude whose sides are tangent to a cycloid?

Problem 1999

From a point $M$ of the curve $y^{2}=ce^{x/a}+4a(x+a)$ , a normal $MP$ is drawn, meeting the axis $Ox$ at a point $P$ . Prove that the midpoint of $MP$ lies on the parabola $y^{2}=ax$ .

Problem 2000

What is the locus of the points from which two mutually perpendicular normals can be drawn to a parabola?

Problem 2001

Under an inversion, a point with polar coordinates $r$ and $\varphi$ is taken to a point with polar coordinates $\dfrac {a^{2}}{r}$ and $\varphi$ , where $a$ is a constant. Prove that an inversion is a conformal mapping, i.e.that the images of curves intersect at the same angles as the curves themselves.

Problem 2002

The parabola $y=-ax^{2}$ rolls without slipping along the parabola $y=ax^{2}$ . Find the locus of its vertex, assuming that at the initial moment the vertices coincided.

Problem 2003

Prove that during the same motion of the parabola, its focus describes a straight line, namely the directrix of the fixed parabola.

Remark.

This refers to the motion of the preceding problem: the parabola $y=-ax^{2}$ rolls without slipping along the parabola $y=ax^{2}$ , and at the initial moment their vertices coincided.

Problem 2004

Along the ellipse $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$ , another, equal ellipse rolls without slipping. In the initial position their major axes touched at the ends. Find the locus of the center of the rolling ellipse.

Remark.

Remark. In solving problems 2002--2004 it is useful to pay attention to the position of the moving and fixed figures relative to their common tangent.

Section 5.3Convexity, Curvature, and Radius of Curvature[53/68%]

Problem 2005

For what values of $x$ is the curve $y=x^{3}+ax+b$ convex upward?

Problem 2006

Determine the direction of convexity of the curve $x(x^{2}-y^{2})+y^{2}=0$ at the points where $x=\dfrac {3}{2}$ .

Problem 2007

Investigate the nature of the concavity of the curve $x^{4}=y(x^{2}-y^{2})$ at points where $y<0$ .

Problem 2008

Investigate the nature of the convexity of the curve $r\cos^{3}3\varphi = 1$ for $-\dfrac {\pi }{6}<\varphi <\dfrac {\pi }{6}$ .

Problem 2009

Show that the curve $x^{4}+y^{4}=x^{2}+y^{2}$ is turned with its convex side toward the origin at the points where it meets the axes, and with its concave side at the points where it meets the bisectors of the angles.

Problem 2010

Show that the curve $x=\varphi (t)$ , $y=\psi (t)$ is turned with its concave side toward the origin if

(x'y''-x''y')(xy'-x'y) > 0 .

Problem 2011

Starting from the definition of curvature, find the length of an arc of $1^{\circ }$ of the Earth's meridian at the pole and at the equator. The meridian may be taken as an ellipse with semi-axes $a=6378$ km and $b=a-21$ km (the polar semi-axis).

Problem 2012

Find the radius of curvature of the curve $3ay^{2}=2x^{3}$ .

Problem 2013

Find the smallest radius of curvature of the parabola $y^{2}=2px$ .

Problem 2014

Find the radius of curvature of the curve $y=x^{3}$ at the point $(1,1)$ .

Problem 2015

Find the radius of curvature of the curve $x=t^{2}$ , $y=t^{3}$ at the point $(1,1)$ .

Problem 2016

Find the radius of curvature of the curve $y=a\ch \dfrac {x}{\alpha }$ (the catenary) at the point $(0,a)$ .

Problem 2017

Find the radius of curvature of the curve $y=a\ln \cos \dfrac {x}{a}$ .

Problem 2018

Find the radius of curvature of the curve $x^{n}+y^{n}=2a^{n}$ at the point $(a,a)$ .

Problem 2019

Find the radius of curvature of the curve $x=a\cos^{4}t$ , $y=a\sin^{4}t$ at the point $(a,0)$ .

Problem 2020

Find the radius of curvature of the curve $x=a(\cos t+t\sin t)$ , $y=a(\sin t-t\cos t)$ .

Problem 2021

Find the radius of curvature of the curve $x=a(t-\sin t)$ , $y=a(1-\cos t)$ .

Problem 2022

Find the radius of curvature of the curve $r=a(1+\cos \varphi )$ .

Problem 2023

Find the radius of curvature of the curve $r^{2}=a^{2}\cos 2\varphi$ , at $\varphi =0$ .

Problem 2024

Find the radius of curvature of the curve

\begin{cases} x = a(t+\sin t)+b\sin \dfrac {t}{2}, \\ y = a(3+\cos t)+b\cos \dfrac {t}{2}. \end{cases}

Problem 2025

Find the radius of curvature of the curve

\begin{cases} x = ae^{-t}(\cos t-\sin t), \\ y = ae^{-t}(\cos t+\sin t). \end{cases}

Problem 2026

Find the radius of curvature of the curve

\begin{cases} x = a\bigl(3t+\sin t\cos t(2\cos ^{2}t+3)\bigr), \\ y = 2a\cos ^{4}t. \end{cases}

Problem 2027

Find the greatest curvature of the curve $y=\ln x$ .

Problem 2028

Find the greatest curvature of the curve $x=a\cos t$ , $y=b\sin t$ (an ellipse).

Problem 2029

Find the greatest curvature of the curve $y=a\ln \left(1-\dfrac {x^{2}}{a^{2}}\right)$ .

Problem 2030

Find the greatest curvature of the curve $y=a\ch \dfrac {x}{a}$ .

Problem 2031

If, along the normal from a point of a curve, one lays off in the direction of concavity of the curve a length equal to the radius of curvature, one obtains the center of curvature. Prove that its coordinates for the curve $x=\varphi (t)$ , $y=\psi (t)$ are given by the formulas:

X-x = -\frac{y'(x'^{2}+y'^{2})}{x'y''-x''y'}, \qquad Y-y = \frac{x'(x'^{2}+y'^{2})}{x'y''-x''y'}.

Problem 2032

Find the coordinates of the center of curvature of the hyperbola $xy=a^{2}$ at the point $(a,a)$ .

Problem 2033

Find the coordinates of the center of curvature of the strophoid $y^{2}=x^{2}\dfrac {a+x}{a-x}$ .

Problem 2034

Find the coordinates of the center of curvature of the curve $y=x\ln x$ at the point where $y'=0$ .

Problem 2035

Find the coordinates of the center of curvature of the curve $r=a(1+\cos \varphi )$ .

Problem 2036

Find the coordinates of the center of curvature of the curve $r=a\cos^{3}\varphi$ .

Problem 2037

Prove that for points of the Archimedean spiral $r=a\varphi$ , as $\varphi \rightarrow \infty$ , the difference between the radius vector and the radius of curvature tends to zero.

Problem 2038

Prove that under the same conditions (see the preceding problem) the center of curvature moves along a curve that tends to coincide with the circle $r=a$ .

Problem 2039

Prove that for the curves $r^{n}=a^{n}\cos n\varphi$ , the polar normal is $n+1$ times the radius of curvature.

Problem 2040

Prove that for the curves $r^{n}=a^{n}\sin n\varphi$ , the part of the radius vector lying inside the circle of curvature equals $\dfrac {2r}{n+1}$ .

Problem 2041

Prove the equality $R=\dfrac {r\, dr}{dp}$ , where $R$ is the radius of curvature, $r$ is the radius vector, and $p$ is the perpendicular from the pole to the tangent.

Problem 2042

Prove that on an ellipse there exist, in general, three points such that the circles of curvature of the ellipse at these points pass through a given point of the ellipse.

Problem 2043

Prove that the centers of curvature at points of the Archimedean spiral $r=a\varphi$ lying on one ray lie on an ellipse whose dimensions do not depend on the choice of the ray.

Problem 2044

Find the smallest distance from the center of curvature of an ellipse to the center of the ellipse.

Problem 2045

The coordinates of the points of a curve can be expressed in terms of the arc length of the curve: $x=x(s)$ , $y=y(s)$ . Denoting by $\Delta s$ a small increment of $s$ , and by $\Delta l$ the corresponding chord, prove the equality

\Delta l = \Delta s - \frac{1}{24R^{2}}\Delta s^{3}+\ldots ,

where $R$ is the radius of curvature of the curve.

Problem 2046

The point of tangency is taken as the origin, the tangent as the axis $Ox$ , and the axis $Oy$ is directed from the point of tangency toward the center of curvature. Prove that in a neighborhood of the point of tangency, i.e.for small values of $x$ , the following formulas hold:

y_{1} = \frac{x^{2}}{2R}+ax^{3}+\ldots , \qquad y_{2} = \frac{x^{2}}{2R}+\frac{x^{4}}{8R^{3}}+\ldots ,

where $R$ is the radius of curvature at the point $(0,0)$ , $y_{1}$ is the ordinate of points of the curve, and $y_{2}$ is the ordinate of points of the circle of curvature.

Problem 2047

Find a parabola joining the origin $O$ to the point $B(a+b,0)$ so that the arc $OB$ of the parabola, together with the lower half of the circle $(x-a)^{2}+y^{2}=b^{2}$ , forms a smooth curve with continuous curvature.

Problem 2048

Find the parabola $y=ax^{2}+bx+c$ that has, with the sinusoid $y=\sin x$ , at the point $\left(\dfrac {\pi }{2}, 1\right)$ , a common tangent and a common curvature.

Problem 2049

Find the fifth-degree parabola

y = Ax^{5}+Bx^{4}+Cx^{3}+Dx^{2}+Ex+F,

which at the points $(a,h)$ , $(-a,-h)$ is tangent to the lines $y=h$ and $y=-h$ , having contact of the second order.

Problem 2050

Prove that the circle $(x-3a)^{2}+(y-3a)^{2}=8a^{2}$ and the parabola $\sqrt{x}+\sqrt{y}=2\sqrt{a}$ have contact of the third order at the point $(a,a)$ .

Problem 2051

Prove that the osculating circle (circle of curvature) of a second-order curve at the vertex of the curve has contact of the third order.

Problem 2052

Prove that for every point at which a curve is sufficiently smooth, there exists a second-order curve having, with the given curve at the given point, contact of the fourth order.

Problem 2053

Prove that for a cycloid, the second-order curve with contact of the fourth order is always an ellipse.

Problem 2054

Prove that for a logarithmic spiral, the second-order curves having contact of the fourth order with it are ellipses whose major axes make a constant angle with the radius vector of the point of tangency.

Problem 2055

Find the locus of the foci of parabolas having, at a given point, contact of the second order with a given curve.

Problem 2056

Find the locus of the midpoints of the chords common to the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ and its osculating circle.

Problem 2057

Find the locus of the feet of the perpendiculars dropped from the center of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ onto the chords common to the ellipse and its osculating circle.

Section 5.4Evolutes of Curves[23/96%]

Problem 2058

The locus of the centers of curvature of a curve is called the evolute of the given curve. The equation of the evolute for the curve $x=\varphi (t)$ , $y=\psi (t)$ is obtained by eliminating the parameter $t$ from the equations

X-x = -\frac{y'(x'^{2}+y'^{2})}{x'y''-x''y'}, \qquad Y-y = \frac{x'(x'^{2}+y'^{2})}{x'y''-x''y'}.

Find the evolute of the parabola $y^{2}=2px$ .

Problem 2059

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the ellipse $x=a\cos t$ , $y=b\sin t$ .

Problem 2060

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the hyperbola $x=a\ch t$ , $y=b\sh t$ .

Problem 2061

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the astroid $x=a\cos^{3}t$ , $y=a\sin^{3}t$ .

Problem 2062

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the cissoid $y^{2}(2a-x)=x^{3}$ .

Problem 2063

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the hyperbola $xy=a^{2}$ .

Problem 2064

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the catenary $y=a\ch \dfrac {x}{a}$ .

Problem 2065

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the cycloid $x=a(t-\sin t)$ , $y=a(1-\cos t)$ .

Problem 2066

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the curve

x = a\bigl(2t\cos t+(t^{2}-2)\sin t\bigr), \qquad y = a\bigl(2t\sin t-(t^{2}-2)\cos t\bigr).

Problem 2067

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the curve

x = a\cos t+(at+b)\sin t, \qquad y = a\sin t-(at+b)\cos t.

Problem 2068

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the cardioid $r=a(1+\cos \varphi )$ .

Problem 2069

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the hypocycloid

x = a(2\cos t+\cos 2t), \qquad y = a(2\sin t-\sin 2t).

Problem 2070

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the tractrix $x = -a\left(\ln \tg \dfrac {t}{2}+\cos t\right)$ , $y=a\sin t$ .

Problem 2071

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the curve $y=a\ln \cos \dfrac {x}{a}$ .

Problem 2072

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the involute of a circle

x = a(\cos t+t\sin t), \qquad y = a(\sin t-t\cos t).

Problem 2073

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the logarithmic spiral $r=e^{a\varphi }$ .

Problem 2074

Prove the equality

\frac{x'^{2}+y'^{2}}{x'y''-x''y'} = \frac{\xi '^{2}+\eta '^{2}}{\xi '\eta ''-\xi ''\eta '},

where $(x,y)$ is a point of the curve, and $(\xi ,\eta )$ is the corresponding point of the evolute.

Problem 2075

Using the properties of the evolute, find the arc length of one arch of the cycloid $x=a(t-\sin t)$ , $y=a(1-\cos t)$ .

Problem 2076

Using the properties of the evolute, find the arc length of the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ between the points $(a,0)$ and $(0,a)$ .

Problem 2077

Using the properties of the evolute, find the arc length of the cardioid $r=a(1+\cos \varphi )$ .

Problem 2078

Using the properties of the evolute, find the arc length of Neile's parabola $y^{2}=x^{3}$ from $(0,0)$ to $(4,8)$ .

Problem 2079

Using the properties of the evolute, find the arc length of the evolute of the parabola $y^{2}=2px$ from the cusp to the point where the evolute meets the parabola.

Problem 2080

Using the properties of the evolute, find the arc length of the logarithmic spiral $r=e^{a\varphi }$ for $\varphi_{0}<\varphi <\varphi_{0}+2\pi$ .

Remark.

Remark. All the curves listed above (problems 2075--2080) are evolutes of other curves, which are easily found using the solutions of the problems given in the first part of the section.

Section 5.5Envelopes of Curves[23/96%]

Problem 2081

The equation $f(x,y,a)=0$ , for various values of $a$ , represents a family of curves. Sometimes this family has an envelope, i.e.a curve tangent to the curves of the family. The coordinates of the point of tangency of a curve of the family and the envelope satisfy the equations:

f(x,y,a)=0, \qquad \frac{\partial f}{\partial a}=0 .

Eliminating the parameter $a$ from these, we obtain an equation satisfied by all points of the envelope. However, the same equations are also satisfied by the points of the locus of singular points of the curves of the family, i.e.points of the curves where $\dfrac {\partial f}{\partial x}=0$ , $\dfrac {\partial f}{\partial y}=0$ .

Find the envelope of the family of lines $(x-a)^{2}+y^{2}=1$ .

Problem 2082

Find the envelope of the family of lines $(x-a)^{2}+y^{2}=\dfrac {a^{2}}{2}$ .

Problem 2083

Find the envelope of the family of lines $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ , where $a^{2}+b^{2}=1$ .

Problem 2084

Find the envelope of the family of lines $\lambda x^{2}+\mu y^{2}=1$ , where $\lambda +\mu =\lambda \mu$ .

Problem 2085

Find the envelope of the family of lines $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$ , where $a+b=d$ .

Problem 2086

Find the envelope of the family of lines that cut off, from a given angle of magnitude $\omega$ , a triangle of area $2m^{2}$ .

Problem 2087

Find the envelope of the family of lines $bx+ay=ab$ , where $a+b=c$ .

Problem 2088

Find the envelope of the family of lines $bx+ay=ab$ , where $a^{2}+b^{2}=c^{2}$ .

Problem 2089

The vertex of an angle of given magnitude $\alpha$ slides along the axis $Ox$ , and one of its sides passes through the point $(0,h)$ . Find the envelope of the other side.

Problem 2090

A square with side $a$ moves so that two of its sides, $AB$ and $BC$ , pass through the points $(\pm b, 0)$ . Find the envelope of the diagonal $AC$ .

Problem 2091

A line rotates uniformly about a point moving with a given speed along a given line. Find the envelope of the moving line.

Problem 2092

A circle rolls without slipping along a given line. Find the envelope of its diameter.

Problem 2093

On the chords of the circle $x^{2}+y^{2}=a^{2}$ parallel to the axis $Oy$ , circles are constructed having these chords as diameters. Find their envelope.

Problem 2094

Find the envelope of the circles having their centers on a parabola and passing through its vertex.

Problem 2095

Find the envelope of the circles constructed, as diameters, on the chords of a parabola that pass through the focus.

Problem 2096

Find the envelope of the system of circles having their centers on a given ellipse and passing through the center of the ellipse.

Problem 2097

Find the envelope of the circles with centers on the hyperbola $xy=a^{2}$ that are tangent to the axis $Ox$ .

Problem 2098

On the radius vectors of a curve, circles are constructed having them as diameters. Prove that the envelope of these circles is the pedal curve of the curve with respect to the pole. (The pedal curve of a curve with respect to a given point is the locus of the feet of perpendiculars dropped from the given point onto the tangents to the given curve; see problem 1981.)

Problem 2099

The circle $x^{2}+y^{2}=r^{2}$ is the envelope of the family $\dfrac {x}{a}+\dfrac {y}{b}=1$ . What is the relation between $a$ and $b$ ?

Problem 2100

The variables $z$ and $t$ are related by the equation $\left(\dfrac {z}{a}\right)^{m}+\left(\dfrac {t}{b}\right)^{m}=1$ . Find the envelope of the curves $\left(\dfrac {x}{z}\right)^{n}+\left(\dfrac {y}{t}\right)^{m}=1$ .

Problem 2101

A luminous point has coordinates $(a,0)$ . Find the envelope of the rays reflected from the circle $x^{2}+y^{2}=a^{2}$ . (A caustic.)

Problem 2102

Parallel rays fall on a spherical mirror. Find the envelope of the reflected rays.

Problem 2103

A luminous point has coordinates $(a,0)$ . Find the envelope of the rays reflected from the parabola $y^{2}=2px$ .

Section 5.6Plotting Curves[124/96%]

Problem 2104

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $x^{2}+xy+y^{2}=3$ .

Problem 2105

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $x^{4}+y^{4}=x^{2}+y^{2}$ .

Problem 2106

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $(x^{2}+y^{2})^{2}=xy$ .

Problem 2107

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $x^{4}+4x^{2}y^{2}-6a^{2}x^{2}+a^{4}=0$ .

Problem 2108

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $y=\sin \dfrac {1}{x}$ .

Problem 2109

Find the vertices, i.e.the points where the tangent is parallel to one of the coordinate axes and where there is, moreover, no point of inflection, of the curve $y=\sin x^{2}$ .

Problem 2110

Find the horizontal distance between successive vertices of the curve $y=e^{-ax}\cos bx$ .

Problem 2111

Find the points of inflection of the curve $y=x^{4}-6x^{2}$ .

Problem 2112

Find the points of inflection of the curve $x^{3}+y^{3}=a^{3}$ .

Problem 2113

Find the points of inflection of the curve $x^{3}+y^{3}=x^{2}$ .

Problem 2114

Find the points of inflection of the curve $(a^{2}+x^{2})y=ax^{2}$ .

Problem 2115

Find the points of inflection of the curve $y=1+\sqrt[3]{x^{5}}$ .

Problem 2116

Find the points of inflection of the curve $y=x^{4}e^{-x}$ .

Problem 2117

Find the points of inflection of the curve $y=e^{1/x}$ .

Problem 2118

Find the points of inflection of the curve $y=x^{2}\ln x$ .

Problem 2119

Find the points of inflection of the curve $y=\tg x$ .

Problem 2120

Find the points of inflection of the curve $y=\sin^{4}x+\cos^{4}x$ .

Problem 2121

Find the points of inflection of the curve $r\cos^{3}\varphi =1$ .

Problem 2122

Find the points of inflection of the curve $r=a(\tg \varphi -1)$ .

Problem 2123

In investigating the shape of a curve near a given point $(a,b)$ of it, an expansion in a Taylor series is useful. If $f(x,y)=0$ is the equation of the curve, then, expanding the left-hand side in powers of the differences $x-a=\xi$ , $y-b=\eta$ , we obtain, by virtue of the equality $f(a,b)=0$ :

p\xi +q\eta +\frac{1}{2}(r\xi ^{2}+2s\xi \eta +t\eta ^{2})+\ldots = 0 ,

where $p$ , $q$ , $r$ , $s$ , and $t$ denote, respectively, the values of the derivatives $\dfrac {\partial f}{\partial x}$ , $\dfrac {\partial f}{\partial y}$ , $\dfrac {\partial^{2}f}{\partial x^{2}}$ , $\dfrac {\partial^{2}f}{\partial x\, \partial y}$ , and $\dfrac {\partial^{2}f}{\partial y^{2}}$ at $x=a$ , $y=b$ .

If $p^{2}+q^{2}>0$ , then for small values of $\left|\xi \right|$ and $\left|\eta \right|$ the curve is close to the line $p\xi +q\eta =0$ , i.e.to the line $p(x-a)+q(y-b)=0$ . This line is the tangent to the curve at the point $(a,b)$ . If, however, $p=0$ and $q=0$ , i.e.if at the given point both derivatives $\dfrac {\partial f}{\partial x}$ and $\dfrac {\partial f}{\partial y}$ vanish, then the equation above becomes:

\frac{1}{2}\left[r\xi ^{2}+2s\xi \eta +t\eta ^{2}\right]+\ldots = 0 .

The point $(a,b)$ is a singular point of the curve.

If, moreover, $s^{2}-rt>0$ , then the point $(a,b)$ is a double point. Two branches of the curve intersect there. The slopes of the tangents to these branches equal the roots of the quadratic equation $t\sigma^{2}+2s\sigma +r=0$ .

If $s^{2}-rt<0$ , then the point $(a,b)$ is an isolated point: sufficiently near it there are no points of the curve. If $s^{2}-rt=0$ , then to fully investigate the nature of the singular point one must take into account further terms of the Taylor expansion. Usually in this case $(a,b)$ is a cusp of the curve.

Investigate the singular points of the algebraic curve $y^{2}=x^{3}$ .

Problem 2124

Investigate the singular points of the algebraic curve $y^{2}=bx^{3}+ax^{2}$ .

Problem 2125

Investigate the singular points of the algebraic curve $y^{2}=ax^{2}+bx^{5}$ .

Problem 2126

Investigate the singular points of the algebraic curve $x^{3}+y^{3}-xy=0$ .

Problem 2127

Investigate the singular points of the algebraic curve $x^{4}-2ax^{2}y+2ay^{3}+a^{2}y^{2}=0$ ; $a\neq 0$ .

Problem 2128

Investigate the singular points of the algebraic curve $x^{6}-2a^{2}x^{3}y-b^{3}y^{3}=0$ .

Problem 2129

Investigate the singular points of the algebraic curve $2y^{3}x-y^{4}=x(y-x)^{2}$ .

Problem 2130

Investigate the singular points of the algebraic curve $x^{4}+y^{4}=x^{2}+y^{2}$ .

Problem 2131

For what relation between $a$ and $b$ does the curve

y^{2}=x^{3}+ax+b

have a double point?

Problem 2132

Prove that for $4a^{3}+27b^{2}<0$ the curve $y^{2}=x^{3}+ax+b$ consists of two separate parts, while for $4a^{3}+27b^{2}>0$ it is a single connected curve.

Problem 2133

In studying transcendental curves, Taylor's formula is not always applicable, so transcendental curves have singular points of kinds not found on algebraic curves (such are, for example, points of cessation and corner points). Investigate the singular points of the transcendental curve $y=x^{x}$ .

Problem 2134

Investigate the singular points of the transcendental curve $y=x\ln x$ .

Problem 2135

Investigate the singular points of the transcendental curve $y\ln x=1$ .

Problem 2136

Investigate the singular points of the transcendental curve $y=e^{1/x}$ .

Problem 2137

Investigate the singular points of the transcendental curve $y=\dfrac {1}{1+e^{1/x}}$ .

Problem 2138

Investigate the singular points of the transcendental curve $y=\dfrac {x}{1+e^{1/x}}$ .

Problem 2139

Investigate the singular points of the transcendental curve $y=\arctg \dfrac {1}{x}$ .

Problem 2140

Investigate the singular points of the transcendental curve $y=x\arctg \dfrac {1}{x}$ .

Problem 2141

Investigate the singular points of the transcendental curve $y=\sin \dfrac {1}{x}$ .

Problem 2142

Investigate the singular points of the transcendental curve $y=x\sin \dfrac {1}{x}$ .

Problem 2143

Investigate the singular points of the transcendental curve $y=x^{2}\sin \dfrac {1}{x}$ .

Problem 2144

Investigate the singular points of the transcendental curve $x^{y}=y^{x}$ .

Problem 2145

In finding the asymptotes of a curve one should distinguish two cases — asymptotes parallel to the axis $Oy$ , and asymptotes not parallel to it. The former have equations $x=x_{0}$ , where $x_{0}$ is the value of $x$ at which $y=f(x)$ becomes infinite. A line $y=ax+b$ , not parallel to the axis $Oy$ , will be an asymptote of the curve if, as $x\rightarrow \infty$ , the points of the curve satisfy the equality

y=ax+b+\varepsilon (x), \qquad \text{where } \varepsilon (x)\rightarrow 0 \text{ as } x\rightarrow \infty .

It follows that $a=\lim \limits_{x\rightarrow \infty }\dfrac {y}{x}$ , $b=\lim \limits_{x\rightarrow \infty }(y-ax)$ .

Find the asymptotes of the curve $y=\dfrac {x}{x-1}$ .

Problem 2146

Find the asymptotes of the curve $y=\dfrac {1}{(x-3)^{2}}$ .

Problem 2147

Find the asymptotes of the curve $y=\pm \sqrt{\dfrac {x-1}{x+1}}$ .

Problem 2148

Find the asymptotes of the curve $y=x+\dfrac {1}{x}$ .

Problem 2149

Find the asymptotes of the curve $y=\dfrac {1}{x}+\dfrac {1}{x-1}+\dfrac {1}{x-2}$ .

Problem 2150

Find the asymptotes of the curve $y^{2}=\dfrac {x^{3}+ax^{2}}{x-a}$ .

Problem 2151

Find the asymptotes of the curve $y^{2}=\dfrac {x^{3}-a^{3}}{x+b}$ .

Problem 2152

Find the asymptotes of the curve $y^{3}-x^{3}=x^{2}+y^{2}$ .

Problem 2153

Find the asymptotes of the curve $x^{3}+y^{3}=a^{3}$ .

Problem 2154

Find the asymptotes of the curve $x^{3}+y^{3}=3axy$ .

Problem 2155

Find the asymptotes of the curve $x^{3}+y^{3}+2y-x=0$ .

Problem 2156

Find the asymptotes of the curve $r\varphi =a$ .

Problem 2157

Study the shape of the curve $y^{2}=x^{3}(2-x)$ , first solving the equation of the curve for one of the coordinates.

Problem 2158

Study the shape of the curve $x^{4}+y^{4}=a^{4}$ , first solving the equation of the curve for one of the coordinates.

Problem 2159

Study the shape of the curve $x^{2n}+y^{2n}=a^{2n}$ , where $n$ is a large positive integer, first solving the equation of the curve for one of the coordinates.

Problem 2160

Study the shape of the curve $x^{4}-4x^{2}y^{2}-6x^{2}-4y^{2}=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2161

Study the shape of the curve $x^{4}-6x^{2}y+25y^{2}-16x^{2}=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2162

Study the shape of the curve $y^{2}-x^{4}+x^{6}=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2163

Study the shape of the curve $x^{4}+x^{2}y^{2}-18x^{2}y+9y^{2}=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2164

Study the shape of the curve $x^{4}+y^{4}-3x^{3}-4x^{2}=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2165

Study the shape of the curve $y^{4}-2xy^{2}-3x^{2}+x^{4}=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2166

Study the shape of the curve $x^{4}+y^{4}-6y^{3}+8x^{2}y=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2167

Study the shape of the curve $x^{4}+y^{4}-8x^{2}-10y^{2}+16=0$ , first solving the equation of the curve for one of the coordinates.

Problem 2168

Study the shape of the closed curve $x^{4}+y^{4}=2xy$ , by passing to polar coordinates.

Problem 2169

Study the shape of the closed curve $(x^{2}+y^{2}-6x)^{2}=x^{2}+y^{2}$ (the limaçon of Pascal), by passing to polar coordinates.

Problem 2170

Study the shape of the closed curve $(x^{2}+y^{2}-4x)^{2}=16(x^{2}+y^{2})$ (a cardioid), by passing to polar coordinates.

Problem 2171

Study the shape of the closed curve $x^{4}+y^{4}=8xy^{2}$ , by passing to polar coordinates.

Problem 2172

Study the shape of the closed curve $x^{4}+y^{4}=x^{2}+y^{2}$ , by passing to polar coordinates.

Problem 2173

Study the shape of the closed curve $(x^{2}+y^{2})^{3}=27x^{2}y^{2}$ , by passing to polar coordinates.

Problem 2174

Study the shape of the closed curve $(x^{2}+y^{2})^{2}=xy$ , by passing to polar coordinates.

Problem 2175

Plot the curve (take $a>0$ ) $y^{2}=x^{3}-2x^{2}+x$ .

Problem 2176

Plot the curve (take $a>0$ ) $4y^{2}=4x^{2}y+x^{5}$ .

Problem 2177

Plot the curve (take $a>0$ ) $x^{5}+5ax^{4}-16a^{3}y^{2}=0$ .

Problem 2178

Plot the curve (take $a>0$ ) $y^{3}-x^{2}y+x^{5}=0$ .

Problem 2179

Plot the curve (take $a>0$ ) $y^{5}+x^{4}=xy^{2}$ .

Problem 2180

Plot the curve (take $a>0$ ) $x^{6}+2x^{3}y-y^{3}=0$ .

Problem 2181

Plot the curve (take $a>0$ ) $x^{4}+2y^{3}=4x^{2}y$ .

Problem 2182

Plot the curve (take $a>0$ ) $y^{3}-x^{3}+y-2x=0$ .

Problem 2183

Plot the curve (take $a>0$ ) $(x^{2}-y^{2})^{2}=2x$ .

Problem 2184

Plot the curve (take $a>0$ ) $x^{2}(x-y)^{2}+y=0$ .

Problem 2185

Plot the curve (take $a>0$ ) $x^{3}y^{3}=x-y$ .

Problem 2186

Plot the curve (take $a>0$ ) $x^{2}y^{2}+x-2y=0$ .

Problem 2187

Plot the curve (take $a>0$ ) $(2x+y)^{2}(x+y)=x$ .

Problem 2188

Plot the curve (take $a>0$ ) $(x^{2}-y^{2})(x-y)=1$ .

Problem 2189

Plot the curve (take $a>0$ ) $3x^{4}-y^{4}+2x^{2}y^{2}+2x=0$ .

Problem 2190

Plot the curve (take $a>0$ ) $xy(x-y)+x+y=0$ .

Problem 2191

Plot the curve (take $a>0$ ) $x^{2}y^{2}+y=1$ .

Problem 2192

Plot the curve (take $a>0$ ) $xy(x^{2}-y^{2})+1=0$ .

Problem 2193

Plot the curve (take $a>0$ ) $xy(x^{2}+y^{2})-2x^{2}y^{2}=12$ .

Problem 2194

Plot the curve (take $a>0$ ) $x(x^{2}+y^{2})=a(x^{2}-y^{2})$ .

Problem 2195

Plot the curve (take $a>0$ ) $(x^{2}-y^{2})^{2}=4xy$ .

Problem 2196

Plot the curve (take $a>0$ ) $x^{2}y^{2}+y^{4}=4x^{2}$ .

Problem 2197

Plot the curve (take $a>0$ ) $x^{4}-y^{4}+xy=0$ .

Problem 2198

Plot the curve (take $a>0$ ) $x^{2}(x^{2}+y^{2})=4(x-y)^{2}$ .

Problem 2199

Plot the curve (take $a>0$ ) $x^{4}-y^{4}=4y^{2}-x^{2}$ .

Problem 2200

Plot the curve (take $a>0$ ) $x^{2}y^{2}=(y+1)^{2}(4-y^{2})$ .

Problem 2201

Plot the curve (take $a>0$ ) $(x^{2}-1)y^{2}=x^{4}-4x^{2}$ .

Problem 2202

Plot the curve (take $a>0$ ) $x^{2}(y-2)^{2}+2xy=y^{2}$ .

Problem 2203

Plot the curve (take $a>0$ ) $x^{4}-y^{4}=x^{2}-2y^{2}$ .

Problem 2204

Plot the curve (take $a>0$ ) $(x+1)(x+2)y^{2}=x^{2}$ .

Problem 2205

Plot the curve (take $a>0$ ) $xy(x+y)+x^{2}=2y^{2}$ .

Problem 2206

Plot the curve (take $a>0$ ) $(2a-x)y^{2}=x^{3}$ .

Problem 2207

Plot the curve (take $a>0$ ) $x^{3}+y^{3}=3x^{2}$ .

Problem 2208

Plot the curve (take $a>0$ ) $x^{4}-2x^{2}=y^{3}(x-1)$ .

Problem 2209

Plot the curve (take $a>0$ ) $x^{4}-y^{4}=4x^{2}y$ .

Problem 2210

Plot the curve (take $a>0$ ) $x^{3}-2x^{2}y-y^{2}=0$ .

Problem 2211

Plot the curve (take $a>0$ ) $x^{4}-2x^{2}y^{2}+y^{3}=0$ .

Problem 2212

Plot the curve (take $a>0$ ) $x^{2}y^{2}=x^{2}+y^{2}$ .

Problem 2213

Plot the curve (take $a>0$ ) $x^{5}+y^{5}=xy^{2}$ .

Problem 2214

Plot the curve (take $a>0$ ) $x^{4}+2x^{2}y-xy^{2}+y^{2}=0$ .

Problem 2215

Study the curve whose equations are given in parametric form: $x=\dfrac {2+t^{2}}{1+t^{2}}$ , $y=t-\dfrac {t}{1+t^{2}}$ .

Problem 2216

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}+1}{4(1-t)}$ , $y=\dfrac {t}{t+1}$ .

Problem 2217

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}+1}{t^{2}-1}$ , $y=\dfrac {t}{t^{4}+1}$ .

Problem 2218

Study the curve whose equations are given in parametric form: $x=\dfrac {t}{1-t^{2}}$ , $y=\dfrac {t(1-2t^{2})}{1-t^{2}}$ .

Problem 2219

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}}{t^{2}-1}$ , $y=\dfrac {t^{2}+1}{t+2}$ .

Problem 2220

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}-3}{t^{2}+1}$ , $y=\dfrac {t(t^{2}-3)}{t^{2}+1}$ .

Problem 2221

Study the curve whose equations are given in parametric form: $x=\dfrac {t-t^{2}}{1+t^{2}}$ , $y=\dfrac {t^{2}-t^{3}}{1+t^{2}}$ .

Problem 2222

Study the curve whose equations are given in parametric form: $x=\dfrac {t^{2}}{t-1}$ , $y=\dfrac {t}{t^{2}-1}$ .

Problem 2223

Study the curve whose equations are given in parametric form: $x=\dfrac {(t+2)^{2}}{t+1}$ , $y=\dfrac {(t-2)^{2}}{t-1}$ .

Problem 2224

Study the curve whose equations are given in parametric form: $x=\dfrac {2t-1}{t^{3}(t-1)}$ , $y=\dfrac {2t-1}{t^{2}(t-1)}$ .

Problem 2225

Study the curve $y^{2}=a^{2}\sin \dfrac {y}{x}$ .

Problem 2226

Study the curve $y^{2}=a^{2}\cos \dfrac {y}{x}$ .

Problem 2227

Investigate the curve $x^{y}=y^{x}$ , noting that it splits into the line $y=x$ and the curve

x = (1+\sigma )^{\frac{1}{\sigma }}, \qquad y = (1+\sigma )^{\frac{1}{\sigma }+1}.

Section 5.7Space Curves: The Tangent Line and the Normal Plane[39/56%]

Problem 2228

Show that the curve

x = at^{2}+bt+c, \qquad y = a_{1}t^{2}+b_{1}t+c_{1}, \qquad z = a_{2}t^{2}+b_{2}t+c_{2}

lies in some plane, and find the equation of this plane.

Problem 2229

For what relation among the coefficients $a$ , $b$ , $c$ , $a_{1}$ , $b_{1}$ , $c_{1}$ , $a_{2}$ , $b_{2}$ , $c_{2}$ does the curve

x = a\varphi (t)+b\psi (t)+c\omega (t)+d,

y = a_{1}\varphi (t)+b_{1}\psi (t)+c_{1}\omega (t)+d_{1},

z = a_{2}\varphi (t)+b_{2}\psi (t)+c_{2}\omega (t)+d_{2}

lie in a plane?

Problem 2230

Show that the curve

x = \sin 2\varphi , \qquad y = 1-\cos 2\varphi , \qquad z = 2\cos \varphi

lies on the surface of a sphere.

Problem 2231

Show the same (that the curve lies on the surface of a sphere) for the curve

x = \frac{t}{1+t^{2}+t^{4}}, \qquad y = \frac{t^{2}}{1+t^{2}+t^{4}}, \qquad z = \frac{t^{3}}{1+t^{2}+t^{4}} .

Problem 2232

Convert to parametric form the equation of Viviani's curve, defined by the intersection of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ and the cylinder $x^{2}+y^{2}=ax$ , by setting $x=a\sin^{2}t$ .

Problem 2233

Prove that the projection of the preceding curve onto the plane $xOz$ is a parabola.

Problem 2234

Find the projection onto the plane $xOy$ of the curve of intersection of the paraboloid $z=x^{2}+y^{2}$ and the plane $x+y+z=1$ .

Problem 2235

Find the projection of the curve

x = e^{t}\sin t, \qquad y = e^{t}\cos t, \qquad z = 2t

onto the plane $xOy$ .

Problem 2236

Prove that the projection of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

onto the plane $yOz$ is a sinusoid.

Problem 2237

Prove that, upon translating the origin to the point $O_{1}(0,0,b\beta )$ and rotating the abscissa and ordinate axes about the axis $O_{1}z_{1}$ through an angle $\beta$ , the equations of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

can be brought to the form

x_{1} = a\cos t_{1}, \qquad y_{1} = a\sin t_{1}, \qquad z_{1} = bt_{1} .

This shows that the helix is capable of sliding along itself.

Problem 2238

Find the equation of the tangent to the curve

x = \frac{t^{4}}{4}, \qquad y = \frac{t^{3}}{3}, \qquad z = \frac{t^{2}}{2} .

Problem 2239

Find the tangent to the preceding curve that is parallel to the plane $x+3y+2z=0$ .

Problem 2240

Find the cosines of the angles with the axes for the tangent to the curve

x = t-\sin t, \qquad y = 1-\cos t, \qquad z = 4\sin \frac{t}{2} .

Problem 2241

For the curve

x = -t\cos t+\sin t, \qquad y = t\sin t+\cos t, \qquad z = t+1

find the points at which the tangent is parallel to the plane $yOz$ or $xOz$ .

Problem 2242

Find the tangent to the curve

x^{2}+y^{2} = 10, \qquad y^{2}+z^{2} = 25

at the point $(1,3,4)$ .

Problem 2243

Find the tangent to the curve

2x^{2}+3y^{2}+z^{2} = 47, \qquad x^{2}+2y^{2} = z

at the point $(-2,1,6)$ .

Problem 2244

Find the tangent to the curve

x^{2}+y^{2} = z, \qquad x = y

at the point $(m,m,2m^{2})$ .

Problem 2245

Find the cosines of the angles with the axes for the tangent to the curve

x^{2} = 2az, \qquad y^{2} = 2bz .

Problem 2246

The same question (find the cosines of the angles with the axes for the tangent) as in the preceding problem, for the circle

x^{2}+y^{2}+z^{2} = a^{2}, \qquad y = mx .

Problem 2247

A tangent is drawn to the curve

y^{2} = 2px, \qquad z^{2} = 2qx

at the point where $x=\dfrac {p+q}{2}$ . Find its length from the point of tangency to the plane $x=0$ .

Problem 2248

Find the angle between the tangent to the curve

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

and the vector drawn from the origin to the point of tangency.

Problem 2249

Find the normal plane to the curve

z = x^{2}+y^{2}, \qquad y = x .

Problem 2250

At the point where $x=\dfrac {p}{2}$ , find the normal plane to the curve

y^{2} = 2px, \qquad x^{2}+z^{2} = a^{2} .

Problem 2251

Prove that the normal planes of the curve

x = a\cos t, \qquad y = a\sin \alpha \sin t, \qquad z = a\cos \alpha \sin t

pass through the line

x = 0, \qquad z+y\tg \alpha = 0 .

Problem 2252

Prove that the curve

x = e^{t}\cos t, \qquad y = e^{t}\sin t, \qquad z = e^{t}

intersects all the generators of the cone $x^{2}+y^{2}=z^{2}$ at one and the same angle.

Problem 2253

For what $a$ does the curve

x = e^{at}\cos t, \qquad y = e^{at}\sin t, \qquad z = e^{at}

intersect all the generators of the cone $x^{2}+y^{2}=z^{2}$ at an angle $\dfrac {\pi }{4}$ ?

Problem 2254

Prove that the curves of intersection of the cylinders $y^{2}+z^{2}=m^{2}$ with the surface $xy=az$ intersect at a right angle all the generators of this surface belonging to one family.

Problem 2255

The curve $x=a\tg t$ , $y=b\cos t$ , $z=b\sin t$ lies on the surface of a paraboloid. Prove that it intersects all the generators of one family at a right angle.

Problem 2256

A curve called the loxodrome is defined by the equation $\varphi =a\ln \tg \left(\dfrac {\pi }{4}-\dfrac {\theta }{2}\right)$ , where $\theta$ is the latitude and $\varphi$ is the longitude of a point on a sphere. Prove that it intersects the meridians of the sphere at an angle whose tangent equals $a$ .

Problem 2257

Prove that stereographic projection gives a conformal image, i.e.that lines in the plane intersect at the same angle as their images on the sphere.

Remark.

Stereographic projection is defined as follows. The topmost point of the sphere $x^{2}+y^{2}+z^{2}=az$ , i.e.the point $A(0,0,a)$ , is joined by a line to an arbitrary point $M(x,y,0)$ of the plane $xOy$ . The point $M_{1}$ , at which the line $AM$ meets the surface of the sphere $x^{2}+y^{2}+z^{2}=az$ , is taken as the image on the sphere of the point $M$ of the plane $xOy$ , and the point $M$ is the image of the point $M_{1}(x_{1},y_{1},z_{1})$ lying on the sphere. Thus stereographic projection gives a one-to-one correspondence between the points of the plane and of the sphere, with the exception of the point $A$ at the top of the sphere. Here

x_{1} = \frac{a^{2}x}{x^{2}+y^{2}+a^{2}}, \qquad y_{1} = \frac{a^{2}y}{x^{2}+y^{2}+a^{2}}, \qquad z_{1} = \frac{a(x^{2}+y^{2})}{x^{2}+y^{2}+a^{2}} .

Problem 2258

Prove that the curve $r=e^{m\varphi }$ , lying in the plane $xOy$ , where the polar axis coincides with the positive part of the axis $Ox$ , is taken by stereographic projection (see problem 2257) into a loxodrome.

Problem 2259

Prove that circles on the sphere are taken by stereographic projection (see problem 2257) into circles in the plane, straight lines being regarded as a special case of circles.

Problem 2260

The central projection of the sphere $x^{2}+y^{2}+z^{2}=2az$ onto the plane $xOy$ is formed as follows. An arbitrary point $M(x,y,0)$ of the plane $xOy$ is joined by a line $AM$ to the center of the sphere $A(0,0,a)$ . The point $M_{1}$ , at which $AM$ meets the sphere, is taken as the image of the point $M$ , and conversely. Prove that this projection is not a conformal image.

Problem 2261

Prove that the tangent to the curve

x^{2} = 3y, \qquad 2xy = 9z

makes a constant angle with some fixed direction.

Problem 2262

The coordinates of the points of a certain curve satisfy the relation

(x^{2}+y^{2}+z^{2}-a^{2})(dx^{2}+dy^{2}+dz^{2}) = (x\, dx+y\, dy+z\, dz)^{2} .

Prove that the tangents to this curve are tangent to the sphere

x^{2}+y^{2}+z^{2} = a^{2} .

Problem 2263

Prove that the tangents to the curve

x = a(\sin t+\cos t), \qquad y = a(\sin t-\cos t), \qquad z = be^{-t}

meet the plane $xOy$ in the circle $x^{2}+y^{2}=4a^{2}$ .

Problem 2264

(a)(b)

Find the curves along which the tangents to the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

meet:

(a)

the plane $xOy$ ;

(b)

the plane $xOz$ .

Problem 2265

Find the spherical indicatrix of tangents of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt .

Remark.

The spherical indicatrix of tangents of a given curve is the locus of the endpoints of the unit vectors, drawn from the origin, that are parallel to these tangents.

Problem 2266

The same question (find the spherical indicatrix of tangents, see problem 2265) for the curve

x = 2t-\sin 2t, \qquad y = \cos 2t, \qquad z = 4\sin t .

Section 5.8Space Curves: The Osculating Plane, the Principal Normal, and the Binormal[44/80%]

Problem 2267

Using the third Frenet formula, show that a curve whose torsion is identically zero lies in a plane.

Problem 2268

Find the osculating plane of the curve

y^{2} = x, \qquad x^{2} = z

at the point $(1,1,1)$ .

Problem 2269

Find the osculating plane of the curve

y = \varphi (x), \qquad z = a\varphi (x)+b .

Problem 2270

Find the osculating plane of the curve

x^{2} = 2az, \qquad y^{2} = 2bz .

Problem 2271

Find the osculating plane of the curve $x=e^{t}$ , $y=e^{-t}$ , $z=t\sqrt{2}$ .

Problem 2272

Find the osculating plane of the curve $x=a\cos t$ , $y=a\sin t$ , $z=bt$ .

Problem 2273

Find the osculating plane of the curve $x=at^{2}+bt+c$ , $y=a_{1}t^{2}+b_{1}t+c_{1}$ , $z=a_{2}t^{2}+b_{2}t+c_{2}$ .

Problem 2274

Prove that the osculating plane of the curve

x^{2} = \varphi '(t), \qquad y^{2} = t^{2}\varphi '(t), \qquad z = \varphi (t),

lying on the conoidal surface $z=\varphi \left(\dfrac {y}{x}\right)$ , coincides with the tangent plane to this surface.

Problem 2275

Find the equations of the principal normal and the binormal to the curve $x=a\cos t$ , $y=a\sin t$ , $z=bt$ .

Problem 2276

Find the equations of the principal normal and the binormal to the curve $x=y^{2}$ , $z=x^{2}$ at the point $(1,1,1)$ .

Problem 2277

Find the equations of the principal normal and the binormal to the curve $x=\dfrac {t^{4}}{3}$ , $y=\dfrac {t^{2}}{3}$ , $z=\dfrac {t^{2}}{2}$ .

Problem 2278

Find the equations of the principal normal and the binormal to the curve $x=\dfrac {t^{2}}{2}$ , $y=\dfrac {2t^{3}}{3}$ , $z=\dfrac {t^{4}}{2}$ at the point $\left(\dfrac {1}{2},\dfrac {2}{3},\dfrac {1}{2}\right)$ .

Problem 2279

Find the equations of the spherical indicatrices of the principal normal and the binormal of the helix (see problems 2265, 2275).

Problem 2280

The curve

x = a\ch t\cos t, \qquad y = a\ch t\sin t, \qquad z = at

lies on the surface of revolution $x^{2}+y^{2}=a^{2}\ch^{2}\dfrac {z}{a}$ , called a catenoid. Prove that at every point of the curve the binormal coincides with the normal to the surface.

Problem 2281

Prove that one of the bisectors of the angles between the tangent and the binormal to the curve

x = 3t, \qquad y = 3t^{2}, \qquad z = 2t^{3}

has a constant direction.

Problem 2282

A curve has as its projection onto the plane $xOy$ the sinusoid $y=\sin x$ . What relation must the applicates $z$ of its points satisfy so that the principal normals are parallel to the plane $yOz$ ?

Problem 2283

Along the principal normals of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

segments of given length $l$ are laid off. Find the locus of their endpoints.

Problem 2284

Find the surface on which all the principal normals of the helix of the preceding problem lie.

Problem 2285

The same question (find the surface on which the lines lie) as in the preceding problem, for the binormals.

Problem 2286

Find the radius of curvature $\rho$ of the curve

x = t-\sin t, \qquad y = 1-\cos t, \qquad z = 4\sin \frac{t}{2} .

Problem 2287

Find the radius of curvature $\rho$ of the curve $x=a\ch t\cos t$ , $y=a\ch t\sin t$ , $z=at$ .

Problem 2288

Find the radius of curvature $\rho$ of the curve $x=e^{t}$ , $y=e^{-t}$ , $z=t\sqrt{2}$ .

Problem 2289

Find the radius of curvature $\rho$ of the curve $x=e^{t}\cos t$ , $y=e^{t}\sin t$ , $z=e^{t}$ .

Problem 2290

Find the radius of curvature $\rho$ of the curve $x=a\dfrac {\cos t}{\ch t}$ , $y=a\dfrac {\sin t}{\ch t}$ , $z=a(t-\th t)$ .

Problem 2291

Find the radius of curvature $\rho$ of the curve $x^{2}=2az$ , $y^{2}=2bz$ .

Problem 2292

Find the radius of curvature $\rho$ of the curve $x^{2}-y^{2}+z^{2}=1$ , $y^{2}-2x+z=0$ at the point $(1,1,1)$ .

Problem 2293

Find the radius of curvature $\rho$ of the curve $x=3t-t^{3}$ , $y=3t^{2}$ , $z=3t+t^{3}$ .

Problem 2294

Find the radius of curvature $\rho$ of the curve $x=a(1-\cos 2t)$ , $y=a\sin 2t$ , $z=2a\cos t$ .

Problem 2295

Find the radius of curvature $\rho$ of the curve $x=a\cos^{3}t$ , $y=a\sin^{3}t$ , $z=a\cos 2t$ .

Problem 2296

Find the radius of curvature $\rho$ of the curve $x=\sin z-z\cos z$ , $y=\cos z+z\sin z$ .

Problem 2297

Find the torsion of the curve

y^{2} = x, \qquad x^{2} = z .

Problem 2298

Find the torsion of the curve $x=3t-t^{3}$ , $y=3t^{2}$ , $z=3t+t^{3}$ .

Problem 2299

Find the torsion of the curve $x=\sin u-u\cos u$ , $y=\cos u+u\sin u$ , $z=u$ .

Problem 2300

Find the torsion of the curve $x=a(1-\cos t)$ , $y=a\sin 2t$ , $z=2a\cos t$ .

Problem 2301

Find the curvature and the torsion of the curve

y = \frac{x^{2}}{2a}, \qquad z = \frac{x^{3}}{6a^{2}} .

Problem 2302

The same question (find the curvature and the torsion) as in the preceding problem, for the curve

x = 2abt, \qquad y = a^{2}\ln t, \qquad z = b^{2}t^{2} .

Problem 2303

The same question (find the curvature and the torsion) as in the preceding problem, for the curve

x = a\frac{\cos t}{\cos kt}, \qquad y = a\frac{\sin t}{\cos kt}, \qquad z = \frac{a\sqrt{1+k^{2}}}{k\cos kt} .

Problem 2304

Prove the equality of the radius of curvature and the radius of torsion for the curve $x=\ch z$ , $y=\sh z$ .

Problem 2305

Prove that for the moving trihedron of the curve

x = e^{t}\sin t, \qquad y = e^{t}\cos t, \qquad z = e^{t}

each of its edges makes a constant angle with $Oz$ .

Problem 2306

Under what condition does the center of curvature of the helix

x = a\cos t, \qquad y = a\sin t, \qquad z = bt

lie on the same cylinder as the helix itself?

Problem 2307

Prove that for the curve

x = a\ch t\cos t, \qquad y = a\ch t\sin t, \qquad z = at

the segment of the (non-principal!) normal from a point of the curve to the axis $Oz$ equals the radius of torsion.

Problem 2308

A sphere can be drawn through four points of a curve. If these points tend to a single point, then, under appropriate conditions, this sphere tends to a certain limiting sphere, called the osculating sphere. Find its center and radius.

Problem 2309

Find, to an accuracy of higher-order small terms, the approximate equations of the projections of a curve onto the three planes of the moving trihedron near the point of tangency.

Problem 2310

From the center of a sphere, radii are drawn parallel to the principal normals of a closed curve having no singular points. Prove that the locus of their endpoints divides the surface of the sphere into two parts of equal area. (Jacobi.)

Section 5.9Surfaces. Their Equations[24/83%]

Problem 2311

Surfaces traced out by a line passing through a given point $(a,b,c)$ and a given curve are called conical. Prove that their equations can be represented in the form

f\left(\frac{x-a}{z-c}, \frac{y-b}{z-c}\right) = 0 .

Problem 2312

Surfaces formed by the motion of a line parallel to a vector $\overrightarrow {P}(l,m,n)$ and passing through a given curve are called cylindrical. Prove that the general form of the equation of cylindrical surfaces is

f(nx-lz, ny-mz) = 0 .

Problem 2313

Prove that the equation of a surface of revolution about the axis $Oz$ has the form $f(x^{2}+y^{2}, z)=0$ .

Problem 2314

Prove that the equation of a surface of revolution about the line

x = a+lt, \qquad y = b+mt, \qquad z = c+nt

has the form

f(lx+my+nz, R) = 0,

where

R = \bigl(n(y-b)-m(z-c)\bigr)^{2}+\bigl(l(z-c)-n(x-a)\bigr)^{2}+{}

{}+\bigl(m(x-a)-l(y-b)\bigr)^{2} .

Problem 2315

Find the equation of the cylinder circumscribed about the sphere $x^{2}+y^{2}+z^{2}=a^{2}$ , with generators parallel to the line

x = lt, \qquad y = mt, \qquad z = nt .

Problem 2316

Find the equation of the cylinder with generators parallel to the line $x=y=z$ , circumscribed about the ellipsoid $x^{2}+4y^{2}+9z^{2}=1$ .

Problem 2317

Find the equation of the cylindrical surface whose directrix is the curve

z = 0, \qquad x^{2}+y^{2} = ay,

with generators parallel to the vector $\overrightarrow {P}(l,m,n)$ .

Problem 2318

Find the equation of the conical surface with vertex at the point $(-1,0,0)$ and circumscribed about the paraboloid $2y^{2}+z^{2}=4x$ .

Problem 2319

Find the equation of the conical surface with vertex at the point $(a,b,c)$ , whose directrix is the parabola

z = 0, \qquad y^{2} = 2px .

Problem 2320

Find the equation of the conical surface with vertex at the point $(0,0,-c)$ , whose directrix is the lemniscate

z = 0, \qquad (x^{2}+y^{2})^{2} = a^{2}(x^{2}-y) .

Problem 2321

Circumscribe about the paraboloid $x^{2}+y^{2}=2z$ a conical surface with vertex at the point $(0,0,-2)$ .

Problem 2322

Find the equation of the conical surface circumscribed about the surface $xyz=c^{3}$ with vertex at $(0,0,-3c)$ .

Problem 2323

Find the equation of the surface obtained by rotating the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$ about the axis $Ox$ .

Problem 2324

Find the equation of the surface obtained by rotating the hyperbola $\dfrac {x^{2}}{a^{2}}-\dfrac {y^{2}}{b^{2}}=1$ about the axis $Oy$ .

Problem 2325

Find the equation of the surface obtained by rotating the parabola $y^{2}=2px$ about the axis $Ox$ .

Problem 2326

Find the equation of the surface obtained by rotating the circle $(x-a)^{2}+y^{2}=R^{2}$ about the axis $Oy$ .

Problem 2327

Find the equation of the surface obtained by rotating the parabola $y^{2}=2px$ about the axis $Oy$ .

Problem 2328

Find the equation of the surface obtained by rotating the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ about $Ox$ .

Problem 2329

Find the equation of the surface obtained by rotating the same curve (the lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$ , see problem 2328) about the axis $Oy$ .

Problem 2330

The surface obtained by rotating the sinusoid

x = \sin z, \qquad y = 0

about the axis $Oz$ is illuminated by parallel rays making an angle $\dfrac {\pi }{4}$ with $Oz$ . Find the shape of the shadow it casts on the plane $xOy$ .

Problem 2331

The parametric representation of surfaces, introduced by Gauss, consists in giving the coordinates of the points of the surface as functions of two parameters: $x=\varphi (u,v)$ , $y=\psi (u,v)$ , $z=\omega (u,v)$ .

The equation of a sphere can be represented in the form

x = a\sin \theta \cos \varphi , \qquad y = a\sin \theta \sin \varphi , \qquad z = a\cos \theta .

Find a similar form of the equation for an ellipsoid.

Problem 2332

A surface is given by the equations

x = a\cos ^{4}u\cos ^{4}v, \qquad y = a\cos ^{4}u\sin ^{4}v, \qquad z = a\sin ^{4}u .

Find its equation in ordinary form.

Problem 2333

The same question (find the equation in ordinary form) for the surface given by the equations

x = a\cos ^{3}u\cos ^{3}v, \qquad y = a\cos ^{3}u\sin ^{3}v, \qquad z = a\sin ^{3}v .

Problem 2334

The same question (find the equation in ordinary form) for the surface given by the equations

x = a\frac{u^{2}+v^{2}-1}{u^{2}+v^{2}+1}, \qquad y = b\frac{2u}{u^{2}+v^{2}+1}, \qquad z = c\frac{2v}{u^{2}+v^{2}+1} .

Section 5.10Tangent Planes and Normals. Envelopes[48/54%]

Problem 2335

Find the tangent plane to the surface $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at the point $(x_{1},y_{1},z_{1})$ .

Problem 2336

Find the tangent plane to the surface $x^{2}+y^{2}=z^{2}$ at the point $(x_{1},y_{1},z_{1})$ .

Problem 2337

Find the tangent plane to the surface $xy^{2}+z^{3}=12$ at the point $(1,2,2)$ .

Problem 2338

Find the tangent plane to the surface $x^{n}+y^{n}+z^{n}=a^{n}$ at the point $(x_{1},y_{1},z_{1})$ .

Problem 2339

Find the tangent plane to the surface $(x^{2}+y^{2}+z^{2})^{2}=a^{2}(x^{2}-y^{2}+z^{2})$ at the point $(x_{1},y_{1},z_{1})$ .

Problem 2340

The equation of an algebraic surface has the form $f(x,y,z)=0$ , where $f(x,y,z)$ is a polynomial. By introducing additionally powers of an auxiliary variable $t$ , this polynomial can be made homogeneous, after which the equation of the surface takes the form: $F(x,y,z,t)=0$ at $t=1$ . Prove that the equation of the tangent plane to this surface can be written in the form

\frac{\partial F}{\partial x}X+\frac{\partial F}{\partial y}Y+\frac{\partial F}{\partial z}Z+\frac{\partial F}{\partial t} = 0,

where, after differentiating, the variable $t$ is replaced by unity.

Problem 2341

Determine the distance from the origin to the tangent plane of the surface $x\cos \dfrac {2\pi z}{h}=y\sin \dfrac {2\pi z}{h}$ .

Problem 2342

Draw to the surface $xyz=1$ a tangent plane parallel to the plane $x+y+z-3=0$ .

Problem 2343

(a)(b)

Find the lines along which the tangent plane to the surface $xy=az$ at the point $(x_{1},y_{1},z_{1})$ meets the surface:

(a)

the line in the plane $x=x_{1}$ ;

(b)

the line in the plane $y=y_{1}$ .

Problem 2344

Find the equation of the tangent plane to the skew helicoid

x = u\cos v, \qquad y = u\sin v, \qquad z = av .

Problem 2345

Find the equation of the tangent plane to the surface

x = u+v, \qquad y = u^{2}+v^{2}, \qquad z = u^{3}+v^{3}

at the point $(2,2,2)$ .

Problem 2346

Prove that the tangent planes to the surface $xyz=a^{3}$ form, together with the coordinate planes, a tetrahedron of constant volume.

Problem 2347

Prove that the planes tangent to the surface $\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{a}$ cut off, on the coordinate axes, segments whose sum is constant.

Problem 2348

Prove that for the surface

x^{\frac{2}{3}}+y^{\frac{2}{3}}+z^{\frac{2}{3}} = a^{\frac{2}{3}}

the sum of the squares of the segments cut off on the axes by the tangent plane is a constant quantity.

Problem 2349

Prove that the planes tangent to the surface $z=xf\left(\dfrac {y}{x}\right)$ all pass through one and the same point.

Problem 2350

Prove that the planes tangent to the surface $z=x+f(y-z)$ are all parallel to one and the same line.

Problem 2351

Prove that the segment on the axis $Oz$ cut off by the plane tangent to the surface $z+\sqrt{x^{2}+y^{2}+z^{2}}=x^{n}f\left(\dfrac {y}{x}\right)$ is proportional to the distance from the point of tangency to the origin.

Problem 2352

Find the surface on which the tangents to a helix lie, and prove that the normal to this surface makes a constant angle with the axis $Oz$ .

Problem 2353

Find the points on the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at which the normal makes a given constant angle $\alpha$ with the axis $Ox$ .

Problem 2354

Prove that the point of intersection of the normal to the surface $x^{2}+y^{2}+z^{2}=xf\left(\dfrac {y}{x}\right)$ with the plane $xOy$ is equidistant from the origin and from the foot of the normal.

Problem 2355

The segment of the normal to the surface $z^{2}=2a\sqrt{x^{2}+y^{2}}+b$ , lying between the surface and the plane $xOy$ , is projected onto $xOy$ . Prove that the length of the projection is constant.

Problem 2356

Prove the same property (constancy of the length of the projection, onto the plane $xOy$ , of the segment of the normal lying between the surface and this plane) as in the preceding problem, for the normals to the surface

x = v\cos u-\varphi (u)\cos u+\varphi '(u)\sin u,

y = v\sin u-\varphi (u)\sin u-\varphi '(u)\cos u,

z = \sqrt{2v} .

Problem 2357

From a point $M$ on the surface

x = r\cos \varphi , \qquad y = r\sin \varphi , \qquad z^{2} = -r^{2}+f(re^{\varphi })

a normal $MN$ is drawn to a point $N$ in the plane $xOy$ , and a perpendicular $MP$ is dropped onto the plane $xOy$ . Prove that the angle $NOP$ , where $O$ is the origin, equals $\dfrac {\pi }{4}$ .

Problem 2358

Prove that for the surface $z^{2}=a^{2}\arctg \dfrac {y}{x}+f(x^{2}+y^{2})$ , the area of the triangle $NOP$ , obtained in the same way as in the preceding problem, is a constant quantity.

Problem 2359

In the following problems one is required to establish the mutual orthogonality of the given surfaces (so that, if three surfaces are given, one must prove that each of them meets each of the others at a right angle). Prove the orthogonality of the following systems of surfaces:

xy = az^{2}, \qquad x^{2}+y^{2}+z^{2} = b, \qquad z^{2}+2x^{2} = c(z^{2}+2y^{2}) .

Problem 2360

Prove the orthogonality of the following systems of surfaces (see problem 2359):

xyz = a^{3}, \qquad 2z^{2} = x^{2}+y^{2}+f(x^{2}-y^{2}) .

Problem 2361

Prove the orthogonality of the following systems of surfaces (see problem 2359):

xy = \alpha z, \qquad \sqrt{x^{2}+z^{2}}+\sqrt{y^{2}+z^{2}} = \beta , \qquad \sqrt{x^{2}+z^{2}}-\sqrt{y^{2}+z^{2}} = \gamma .

Problem 2362

Prove the orthogonality of the following systems of surfaces (see problem 2359):

x^{2}+y^{2}+z^{2} = ax, \qquad x^{2}+y^{2}+z^{2} = by, \qquad x^{2}+y^{2}+z^{2} = cz .

Problem 2363

Prove the orthogonality of the following systems of surfaces (see problem 2359):

x(x^{2}+y^{2}+z^{2})+a(x^{2}+y^{2}-z^{2}) = 0, \qquad x^{2}+y^{2}+z^{2}+2ax+2by = 0 .

Problem 2364

Through each point $(x,y,z)$ there pass three surfaces of the form $\dfrac {x^{2}}{a^{2}-\lambda }+\dfrac {y^{2}}{b^{2}-\lambda }+\dfrac {z^{2}}{c^{2}-\lambda }=1$ , for $\lambda =\lambda_{1}, \lambda_{2}, \lambda_{3}$ , representing respectively an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets. Prove the orthogonality of these surfaces (see problem 2359).

Problem 2365

From the points of the line

y = x, \qquad z = \frac{\pi }{4},

lying on the surface $y=x\tg z$ , normals are drawn to the surface. Prove that they lie on a hyperbolic paraboloid.

Problem 2366

The same question (prove that the normals, drawn from the points of a given line lying on the surface, to the surface, lie on a hyperbolic paraboloid) as in the preceding problem, for the line

z = h, \qquad by = x\sqrt{a^{2}-h^{2}}

and the surface $b^{2}y^{2}=x^{2}(a^{2}-z^{2})$ .

Problem 2367

Find the locus of the projections of the center of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ onto its tangent planes.

Problem 2368

Along the ellipsoid of the preceding problem (with $a>b>c$ ), another, equal ellipsoid rolls without slipping or spinning, so that at one moment the endpoints of the major semi-axes were in contact. Find the locus of the center of the (rolling) ellipsoid.

Problem 2369

Prove that the surface $x^{2}y^{2}+x^{2}z^{2}+y^{2}z^{2}=2xyz$ is met by the sphere $x^{2}+y^{2}+z^{2}=1$ in four circles.

Problem 2370

Find the envelope of the following family of surfaces: spheres $(x-a)^{2}+y^{2}+z^{2}=1$ .

Remark.

Surfaces $f(x,y,z,a)=0$ , whose equations contain a coefficient $a$ varying from one surface to another, form a one-parameter family of surfaces. If two infinitely close surfaces meet in a certain curve, then the coordinates of the points of this curve, called the characteristic curve, satisfy the equations:

f(x,y,z,a) = 0, \qquad \frac{\partial f}{\partial a} = 0 .

Eliminating the parameter $a$ from these equations, we find an equation $F(x,y,z)=0$ . If the given family has an envelope surface, then it lies entirely on the surface obtained.

Similarly, the equation $f(x,y,z,a,b)=0$ represents a two-parameter family of surfaces. The limiting position of the point of intersection of three nearby surfaces

f(x,y,z,a,b) = 0, \qquad f(x,y,z,a+\Delta a,b) = 0, \qquad f(x,y,z,a,b+\Delta b) = 0

as $\Delta a\to 0$ and $\Delta b\to 0$ , i.e.the characteristic point of the family of surfaces, satisfies the equations:

f(x,y,z,a,b) = 0, \qquad \frac{\partial f}{\partial a} = 0, \qquad \frac{\partial f}{\partial b} = 0 .

Eliminating the parameters $a$ and $b$ from these, we obtain the equation of a surface $F(x,y,z)=0$ . If the given family has an envelope surface, then all its points satisfy the equation obtained; but other points may satisfy it as well.

Problem 2371

Find the envelope of the following family of surfaces (see problem 2370): planes passing through the point $(\sqrt{2},0,0)$ and at a distance equal to unity from the origin.

Problem 2372

Find the envelope of the spheres

(x-lt)^{2}+(y-mt)^{2}+(z-nt)^{2} = a^{2},

where $t$ is a variable parameter.

Problem 2373

Find the envelope of the spheres whose great circles lie on the paraboloid $z=x^{2}+y^{2}$ .

Problem 2374

The same question (find the envelope of the spheres whose great circles lie on the paraboloid) as in the preceding problem, for the paraboloid $2z=\dfrac {x^{2}}{p}+\dfrac {y^{2}}{q}$ .

Problem 2375

Find the envelope of the spheres of radius $a$ whose centers lie on the circle

x^{2}+y^{2} = r^{2}, \qquad z = 0 .

Problem 2376

Find the envelope of the planes tangent to the parabolas

y^{2} = 2x, \quad z = 0 \qquad \text{and} \qquad y^{2} = 2z, \quad x = 0 .

Problem 2377

Find the envelope of the spheres

(x-lt)^{2}+(y-mt)^{2}+(z-nt)^{2} = p^{2}t^{2},

where $t$ is a variable parameter.

Problem 2378

Find the envelope of the ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ subject to the condition $a^{2}+b^{2}+c^{2}=1$ .

Problem 2379

The same (find the envelope of the ellipsoids $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ , see problem 2378) subject to the condition $a+b+c=1$ .

Problem 2380

Find the envelope of the planes that, together with the coordinate planes, form a tetrahedron of constant volume.

Problem 2381

Find the envelope of the planes for which the segments $a$ , $b$ , $c$ on the axes are related by the equation $a^{n}+b^{n}+c^{n}=1$ .

Problem 2382

Show that the circles

y = tx, \qquad x^{2}+y^{2}+z^{2}-2f(t)x-a^{2} = 0

are the characteristics of the family of spheres

x^{2}+y^{2}+z^{2}-2\bigl(f(t)+tf'(t)\bigr)x-2f'(t)y-a^{2} = 0 .

Section 5.11Lines on Surfaces and Curvature of Surfaces[25/52%]

Problem 2383

Find the principal radii of curvature of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ at the point $(0,0,c)$ .

Problem 2384

Find the principal radii of curvature of the paraboloid $\dfrac {x^{2}}{p}+\dfrac {y^{2}}{q}=2z$ at the point $(0,0,0)$ .

Problem 2385

Find the principal radii of curvature of the helicoid $x=u\cos v$ , $y=u\sin v$ , $z=mv$ .

Problem 2386

Find the principal radii of curvature of the paraboloid $xy=az$ .

Problem 2387

Find the principal radii of curvature of the surface $e^{z}\cos x=\cos y$ .

Problem 2388

Find the principal radii of curvature of the surface $\sin z=\sh x\cdot \sh y$ .

Problem 2389

Find the principal radii of curvature of the catenoid $x^{2}+y^{2}=a^{2}\ch^{2}\dfrac {z}{a}$ .

Problem 2390

Prove that if $\rho_{1}$ and $\rho_{2}$ are the principal radii of curvature at some point of an ellipsoid, then on the curve of intersection of the ellipsoid with a concentric sphere, $\sqrt[4]{\rho_{1}\rho_{2}}=m(\rho_{1}+\rho_{2})$ , where $m$ is a constant.

Problem 2391

An umbilical point, or point of rounding, is a point of a surface at which the principal radii of curvature are equal: $\rho_{1}=\rho_{2}$ . Find the umbilical points of the ellipsoid

\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} = 1 .

Problem 2392

The same (find the umbilical points, see problem 2391) for the ellipsoid

x^{2}+2y^{2}+2z^{2}+2xy-2x-4y-4z+2 = 0 .

Problem 2393

Find the total (Gaussian) curvature of the paraboloid

\frac{x^{2}}{p}+\frac{y^{2}}{q} = 2z .

Problem 2394

Prove that the mean curvature of the surface $z=\ln \cos x-\ln \cos y$ equals zero.

Problem 2395

Prove that the total (Gaussian) curvature of a paraboloid is constant at the points of its intersection with the corresponding elliptic cylinder.

Problem 2396

Find the total and mean curvatures of the surface of revolution $z=f(\rho )$ , where $\rho =\sqrt{x^{2}+y^{2}}$ .

Problem 2397

Prove that the surface of revolution of the tractrix

\frac{x+\sqrt{a^{2}-y^{2}}}{a} = \ln \frac{a+\sqrt{a^{2}-y^{2}}}{y}

about the axis $Ox$ has constant negative curvature, equal to $-\dfrac {1}{a^{2}}$ .

Remark.

Remark. It follows that on this surface the geometry of Lobachevsky holds.

Problem 2398

Through the vertex of the paraboloid of revolution $y^{2}+z^{2}=2px$ , curves are drawn on its surface. Find the locus of their centers of curvature.

Problem 2399

Through a point on a surface, $n$ normal sections are drawn, the angle between consecutive sections being $\dfrac {2\pi }{n}$ . Prove that the arithmetic mean of the curvatures of the sections does not depend on either the number $n$ or the direction of the first section ( $n\geqslant 2$ ).

Problem 2400

A curve on a surface that makes, at each of its points, the greatest angle with the plane $xOy$ possible at that point on the surface is called a line of steepest descent. Prove that the tangent to it, the normal to the surface, and the line through the given point parallel to $Oz$ , lie in one plane.

Problem 2401

Prove that the projections onto the plane $xOy$ of the contour lines of a surface meet the projections of the lines of steepest descent at a right angle.

Problem 2402

Prove that the curves whose projections onto the plane $xOy$ have the equation $x^{a^{2}}=Cy^{b^{2}}$ , where $C$ is a constant, are the lines of steepest descent for the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$ .

Problem 2403

Two surfaces

\frac{x^{2}}{a^{2}+\lambda }+\frac{y^{2}}{b^{2}+\lambda }+\frac{z^{2}}{c^{2}+\lambda } = 1,

whose equations differ in the value of $\lambda$ , are called confocal. Prove that the curves of intersection of an ellipsoid with the hyperboloids of one and of two sheets confocal to it are lines of curvature for the ellipsoid.

Problem 2404

Prove that for the helicoid of problem 2385, along each line of curvature one of the quantities $\ln \left(u+\sqrt{u^{2}+m^{2}}\right)\pm v$ remains constant.

Problem 2405

A curve on a surface whose principal normal everywhere coincides with the normal to the surface is called a geodesic. Show that on the surface of a sphere, an arc of a great circle is a geodesic.

Problem 2406

Show that on the surface

x = a\cos u\cos v, \qquad y = a\sin u\cos v,

z = a\ln \tg \left(\frac{v}{2}+\frac{\pi }{4}\right)-a\sin v

the points of a geodesic satisfy the condition

\frac{du}{dv} = \frac{c\sin v}{\cos ^{2}v\sqrt{a^{2}\cos ^{2}v-c^{2}}},

where $c=\mathrm{const}$ .

Problem 2407

Find the analogous condition for the geodesics of the surface of revolution

x = r\cos \varphi , \qquad y = r\sin \varphi , \qquad z = f(r) .