The Tangent and the Normal

Show that the curves

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

Find the angle at which the curve

intersects the axis $Oy$.

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

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

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

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

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 $\dfrac {3\pi }{4}$.

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 $n\varphi +\dfrac {\pi }{2}$.

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

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 $\varphi \rightarrow \infty$.

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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 $xy$.

Prove that the length of the tangent to the tractrix

equals $a$.

Prove that all normals to the curve

are equidistant from the origin.

Prove that the segment of the normal to the curve

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

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 $2a$.

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

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

The same for the hypocycloid

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

between the coordinate axes.

Find the segment of the normal to the curve

lying between the coordinate axes.

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

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

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.

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

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 $OM : MN$.

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.

Prove that the tangent to the curve

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

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

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 $\pi$.

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 $\mu$.

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.

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.

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.)

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.

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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

Prove that the curves

intersect at an angle $\alpha$.

Prove that the curve

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

Prove that the circles

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

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.

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

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.

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.

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

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

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

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 $(0,0)$.

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

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).

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

Find the pedal curve of the involute of a circle:

with respect to the origin.

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.

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 $M_{1}$.

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?

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

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

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

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

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

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

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 $y^{2}=ax$.

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

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.

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.

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

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.