Second-Order Curves in Simplest Form

Find the equation of the parabola passing through the point $(6,9)$, with vertex at the origin and symmetric with respect to the axis $Oy$.

The axis $Ox$ is the axis of symmetry of a parabola with vertex at the origin. Find the equation of this parabola, given that it passes through the point $(2,2)$.

The parabolic mirror of the telescope reflector at Simeiz had a focal length of $5{,}4$ m and a diameter of $1{,}02$ m. Find the depth of the parabolic concavity of the mirror.

The mirror of an automobile headlight has, in cross-section, the shape of a parabola. The diameter of the mirror is $20$ cm, its depth is $10$ cm. Find the position of the focus of the mirror.

Prove that all parabolas $y^{2}=2px$ are geometrically similar.

On the parabola $y^{2}=24x$ a point is taken at a distance of 14 units from the focus. Determine its distance from the vertex.

Prove that the product of the ordinates of the endpoints of any chord passing through the focus of the parabola $y^{2}=2px$ is a constant quantity.

On the parabola $y^{2}=2px$ find a point such that its distances from the focus and from the vertex are in the ratio $7:8$.

The parabola $y^{2}=4-4x$ and the line $x+y=1$ are given. Through one of their points of intersection, draw another parabola with the same axis and focus.

Set up the equation of the parabola with axis of symmetry $Ox$, passing through the points $(a,0)$, $(0,b)$. For what relation between $a$ and $b$ will the focus of the parabola be at the origin?

The parabolas $y^{2}=4x+8$ and $4x+y^{2}-4y-8=0$ are given. Find the equation of their common chord and its point of intersection with the line joining the foci.

Three points $(-5,2)$, $(1,-4)$ and $(5,12)$ are given, lying on a parabola whose axis is parallel to the axis $Oy$. Find the vertex, focus, and directrix of this parabola.

Through three points $(-2,1)$, $(-1,0)$ and $(6,1)$, draw a circle and a parabola with axis parallel to the axis $Oy$. Determine the distance from the center of the circle to the vertex of the parabola.

Prove that the equation $yy_{1}=p(x+x_{1})$ represents the tangent to the parabola $y^{2}=2px$ at the point $(x_{1},y_{1})$.

The parabolas $y^{2}+4x=0$ and $y^{2}=16x+80$ are given. Verify that they are confocal and that the tangents at the points of intersection are mutually perpendicular.

Three points $(-3,2)$, $(-1,1)$ and $(3,5)$ are given; draw through them a parabola with axis parallel to the axis $Oy$, and a circle, and find the angles at which they intersect.

Prove that the tangent to the parabola $y^{2}=2px$ having slope $m$ is represented by the equation $y = mx+\frac{p}{2m}$.

Find the tangent to the parabola $4y=x^{2}$ at the point $(2,1)$.

Through the point $(0,-4)$ draw a tangent to the parabola $4y=x^{2}$.

Through the point $(-4,-1)$ draw a tangent to the parabola $y^{2}=2x$.

Prove that the tangents drawn at the endpoints of a chord of a parabola passing through the focus are mutually perpendicular.

The parabola $y^{2}=2px$ is given. Find the angle between the tangents to it drawn from the point $(-p,p)$.

Find the distance from the focus of the parabola $y^{2}=2px$ to a tangent to it whose angle with the axis equals $\alpha$.

Find the locus of the feet of the perpendiculars dropped from the focus of a parabola onto its tangents.

The parabola $4y+x^{2}-4x-12=0$ and the line $y-2x=0$ are given. Draw the tangent parallel to the given line.

Find the common tangents to the parabola $y^{2}=8x$ and the circle $x^{2}+y^{2}=2$.

Two parabolas are given: $y^{2}=2px$ and $y^{2}=2p(x+a)$. Prove that the segments of the tangent to the first parabola lying inside the second are bisected by the point of tangency.

Find the locus of points from which the parabola is seen at a right angle.

The ordinates of the circle $x^{2}+y^{2}=36$ are decreased by half. Find the equation of the resulting curve.

Find the semi-axes of the ellipse $3x^{2}+5y^{2}-30=0$.

Find the equation of the ellipse passing through the points $(1,4)$ and $(7,2)$ and symmetric with respect to the axes $Ox$ and $Oy$.

Prove that ellipses with the same eccentricity $e$ are geometrically similar, i.e., can be brought into coincidence with each other by suitably enlarging one of them.

The meridian of the terrestrial globe is an ellipse whose flattening, i.e., $\frac{a-b}{a}$, equals $\frac{1}{298}$. Find its eccentricity.

The orbit of the terrestrial globe is an ellipse with semi-axis $a=150 \cdot 10^{6}$ km and eccentricity $e=0{,}017$. Knowing that the Sun is at the focus of this ellipse, find by how much the shortest distance from the Earth to the Sun (January 4) is shorter than the longest (July 3).

Two points are given, lying on an ellipse whose axes are the coordinate axes — $(3;2{,}4)$ and $(4;1{,}8)$. Find this ellipse and the projections of the foci onto the chord joining the given points.

An ellipse and a parabola are given: $y^{2}=18x-0{,}36x^{2}$, $y^{2}=18x$. Find the distance from the focus of the parabola to the nearest focus of the ellipse.

Find the eccentricity of an equilateral (rectangular) hyperbola.

The directrices of a hyperbola divide the distance between the foci into three equal parts. Find its eccentricity.

The foci of an ellipse divide the distance between the directrices into three equal parts. Find its eccentricity.

What will the equation of the equilateral hyperbola $x^{2}-y^{2}=a^{2}$ become if the axes are rotated through an angle $\alpha =-45^{\circ }$?

The distance between the foci of an ellipse equals 2, and the distance between the directrices equals 10. Find the semi-axes.

The eccentricity of a hyperbola equals 2. Find the angle between the asymptotes.

The minor axis of an ellipse is seen from the focus at a right angle. Find the eccentricity.

The hyperbola $x^{2}-y^{2}=8$ is given. Find the confocal ellipse passing through the point $(4,6)$.

The ellipse $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$ is given. Write the equation of the confocal equilateral hyperbola.

Find the general equation of the ellipses and hyperbolas with foci at the points $(\pm c,0)$.

The ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{5}=1$ is given. Find the equation of the hyperbola having its foci at the vertices of the given ellipse, and its vertices at the foci.

The ellipse $\frac{x^{2}}{33}+\frac{y^{2}}{8}=1$ is given. Find the hyperbola confocal with it, having eccentricity equal to $1{,}25$.

Find the ellipse and the hyperbola passing through the point $(6,4)$ and confocal with the hyperbola

The coordinate axes serve as the axes of a hyperbola passing through the point $(6,5)$ and having eccentricity equal to $1{,}5$. Set up its equation.

Find the points at which the hyperbola $\frac{x^{2}}{64}-\frac{y^{2}}{36}=1$ is met by the lines parallel to the asymptotes and passing through the right focus.

Find the distance from the focus of the hyperbola $b^{2}x^{2}-a^{2}y^{2}=a^{2}b^{2}$ to an asymptote.

The polar equation of a second-order curve is $r(5+3\cos \varphi )=16$. Find its equation with respect to its axes of symmetry.

The same, for the curve $r(4+5\cos \varphi )=9$.

Prove the equality

where $\rho_{1}$, $\rho_{2}$, , $\rho_{n}$ are the radius vectors drawn from the focus of a parabola at angles $\frac{2\pi }{n}$ to one another.

Prove the equality

where $r_{1}$ and $r_{2}$ are mutually perpendicular radius vectors drawn from the center of an ellipse.

Prove the equality

where $r_{1}$, $r_{2}$, , $r_{n}$ are radius vectors from the center of an ellipse, making angles $\frac{2\pi }{n}$ with one another.

Find the equations of the diameters of the ellipse $x^{2}+6y^{2}=2$ whose length equals 2.

Prove that under projection, mutually perpendicular diameters of a circle pass into conjugate diameters of an ellipse.

Using the theorem that the area of a projection equals the area of the figure itself multiplied by the cosine of the angle between the plane of the figure and the plane of projection, prove Apollonius's theorem: $a_{1}b_{1}\sin \omega = ab$, where $a_{1}$ and $b_{1}$ are the lengths of conjugate semi-diameters of an ellipse, and $\omega$ is the angle between them.

Starting from the result of problem 222, prove another theorem of Apollonius, according to which $a_{1}^{2}+b_{1}^{2}=a^{2}+b^{2}$.

Find the angle between the equal conjugate diameters of the ellipse $x^{2}+3y^{2}=6$.

Find the lengths of the conjugate diameters of the hyperbola $x^{2}-y^{2}=1$ between which the angle equals $45^{\circ }$.

Find the lengths of the conjugate diameters of the ellipse $x^{2}+15y^{2}=5$ between which the angle equals $150^{\circ }$.

The angle between conjugate diameters of an ellipse equals $120^{\circ }$. One of them is twice the other. Find the eccentricity.

Find the lengths of the conjugate diameters of the ellipse $\frac{x^{2}}{5}+3y^{2}=1$ making an angle of $150^{\circ }$.

The lengths of the conjugate diameters of the ellipse $8x^{2}+17y^{2}=136$ are in the ratio $4:3$. Find the equations of these diameters.

The sum of the lengths of conjugate diameters is 6, the angle between them is 150, and the eccentricity is $e=\frac{3}{5}$. Find the axes of the ellipse.

Find the lengths of the conjugate diameters of the ellipse $3x^{2}+5y^{2}=15$ making the largest angle.

Find the angle between the conjugate diameters of a hyperbola, given the ratio $m$ of their lengths and the eccentricity $e$.

Find the angle between the asymptotes of a hyperbola, given that its conjugate diameters form an angle of $45^{\circ }$ and are in the ratio 2 to 3.

Find the lengths of the conjugate diameters of the hyperbola $9x^{2}-16y^{2}=144$ whose sum is to the sum of the axes as 5 to 2.

Prove that the equation $Axx_{1}+Byy_{1}+C=0$ represents the tangent to the curve $Ax^{2}+By^{2}+C=0$ at the point $(x_{1},y_{1})$.

The ellipse $4x^{2}+y^{2}=8$ and the line $y-2x=0$ are given. Draw the tangents to the ellipse parallel to this line, and find the distance between the tangents.

Through the point $(2,-1)$ draw a tangent to the ellipse $x^{2}+9y^{2}=9$.

Through the point $(3,-6)$ draw a tangent to the hyperbola $4x^{2}-9y^{2}=36$.

Through the point $(1,-2)$ draw a tangent to the hyperbola $x^{2}-y^{2}=1$.

Draw, parallel to the line $2x-3y=0$, tangents to the ellipse $x^{2}+4y^{2}=4$.

Parallel to the line $10x+3y=0$, draw tangents to the hyperbola $4x^{2}-y^{2}=4$.

Find the tangents to the hyperbola $\frac{4x^{2}}{10}-\frac{y^{2}}{6}=1$ at its points of intersection with the line $3x=5y$, and the distance between these tangents.

Given: the ellipse $\frac{x^{2}}{45}+\frac{y^{2}}{36}=1$ and the hyperbola $\frac{x^{2}}{5}-\frac{y^{2}}{4}=1$. Verify that the tangents to them at the points of intersection are mutually perpendicular.

Two ellipses are given: $\frac{x^{2}}{15}+\frac{y^{2}}{9}=1$ and $\frac{x^{2}}{19}+\frac{y^{2}}{3}=1$. Find the tangents to the first ellipse passing through the foci of the second.

Find the tangents to the hyperbola $\frac{x^{2}}{15}+\frac{y^{2}}{21}=1$ passing through a focus of the conjugate hyperbola.

Find the angle between the tangents to the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ drawn from the point $(1,2)$.

Prove that the tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ having slope $m$ is represented by the equation

Prove that the tangent to the hyperbola $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=\sigma$ having slope $m$ is represented by the equation

Prove that the chords joining the points of intersection of the circle $x^{2}+y^{2}=a^{2}+b^{2}$ with the coordinate axes are tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$.

Show that the ellipses $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ are tangent to the lines $\pm x \pm y = c$, if $a^{2}+b^{2}=c^{2}$.

Prove that the locus of points from which the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is seen at a right angle is the circle $x^{2}+y^{2}=a^{2}+b^{2}$.

Show that for the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, the product of the distances of the foci from a tangent equals $b$.