The Ellipse, Hyperbola, and Parabola and Their Basic Properties

Determine the eccentricity of an ellipse, if the distance between the foci is the arithmetic mean of the lengths of the axes.

Compute the eccentricity of a rectangular (equilateral) hyperbola.

Compute the length of the segment of the asymptote of the hyperbola

enclosed between its center and the directrix.

Prove that, if the semi-axis lengths of an ellipse do not coincide, then the rectangular coordinate system in which it is written by an equation of the form (2.1) is determined uniquely up to the direction of the coordinate axes. State and prove analogous statements for the hyperbola and the parabola.

(Focal property of the ellipse.) Two points $F_{1}$ and $F_{2}$ are given, the distance between which is $2 c$. Find the locus of points for which the sum of the distances to the points $F_{1}$ and $F_{2}$ equals $2 a$, under the condition $a > c$.

(Focal property of the hyperbola.) Two points $F_{1}$ and $F_{2}$ are given, at a distance $2 c$ from each other. Find the locus of points for which the absolute value of the difference of the distances to the points $F_{1}$ and $F_{2}$ equals $2 a$, under the condition $c > a$.

Find the locus of points — centers of circles tangent to two fixed circles with centers $O_{1}, O_{2}$ and radii $r_{1}$ and $r_{2}$, respectively, where $r_{2} > r_{1}$. Consider the following cases:

(Directrix property of the parabola.) Find the locus of points equidistant from a point $F$ and a line $d$, situated at a distance $p$ from $F$.

Find the locus of points — centers of circles tangent to a given line $\ell$ and to a circle of radius $r$ with center $O$.

(Directrix property of the ellipse and hyperbola.) A point $F$, a line $d$, and a positive number $e \neq 1$ are given. Find the locus of points for which the ratio of the distance to the point $F$ to the distance to the line $d$ equals $e$.

Find the locus of points dividing, in the ratio $\lambda \neq 1$, the chords of the circle $x^{2}+y^{2} = a^{2}$ parallel to the axis $O y$.

A circle $x^{2}+y^{2} = a^{2}$ is given. Into what does this circle turn if, without changing the abscissas of its points, the ordinates are decreased by a factor of $k$?

Two points are given: $A_{1} = (-a, 0), A_{2} = (a, 0)$. Find the locus of the points of intersection of lines passing through the points $A_{1}$ and $A_{2}$ and cutting off segments on the ordinate axis whose product equals:

(Optical property of the ellipse.) Prove that the tangent to an ellipse at an arbitrary point bisects the angle between one focal radius and the extension of the other.

(Optical property of the hyperbola.) Prove that the tangent to a hyperbola at an arbitrary point bisects the angle between the focal radii.

(Optical property of the parabola.) Prove that the tangent to a parabola at an arbitrary point bisects the angle between the focal radius and the perpendicular dropped from that point to the directrix.

Prove that hyperbolas sharing common foci with a given ellipse intersect the latter at a right angle.

Determine the greatest and least values of the ratio $\left|\mathbf{r}_{1}\right| /\left|\mathbf{r}_{2}\right|$ for the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1,(a > b)$, where $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ are the focal radii of a point of the ellipse to the left and right foci.

Find the locus of points symmetric to the focus of a nondegenerate second-order curve with respect to the tangents to this curve.

Find the locus of the projections of the focus of a second-order curve onto all possible tangents to it.

Let the trajectory $P_{0} P_{1} P_{2} P_{3} \ldots$ of a ray of light inside a mirrored ellipse not pass through the foci $F_{1}$ and $F_{2}$. Prove that:

(Dandelin spheres.) A right circular cone is given, with angle $\alpha$ between the axis of the cone and its generators. A plane $\Gamma$, not containing the vertex of the cone, makes an angle $\beta$ with the axis of the cone, where $\beta > \alpha$. Inscribe in the cone two spheres tangent to the plane $\Gamma$ at the points $F_{1}$ and $F_{2}$. Prove that the curve of intersection of the plane $\Gamma$ with the cone is an ellipse with foci at the points $F_{1}$ and $F_{2}$.

(Dandelin spheres.) A right circular cone is given, with angle $\alpha$ between the axis of the cone and its generators. A plane $\Gamma$, not containing the vertex of the cone, makes an angle $\beta$ with the axis of the cone, where $\beta < \alpha$. Inscribe in the cone two spheres tangent to the plane $\Gamma$ at the points $F_{1}$ and $F_{2}$. Prove that the curve of intersection of the plane $\Gamma$ with the cone is a hyperbola with foci at the points $F_{1}$ and $F_{2}$.

Under the conditions of the two preceding problems, show that the two planes passing through the circles consisting of the points of tangency of the Dandelin spheres with the cone intersect the plane $\Gamma$ along the directrices of the corresponding ellipse (hyperbola).

Problems 207–208: a right circular cone is given, with angle $\alpha$ between the axis of the cone and its generators. A plane $\Gamma$, not containing the vertex of the cone, makes an angle $\beta$ with the axis of the cone, where $\beta > \alpha$ (respectively $\beta < \alpha$). Inscribe in the cone two spheres tangent to the plane $\Gamma$ at the points $F_{1}$ and $F_{2}$; then the curve of intersection of the plane $\Gamma$ with the cone is an ellipse (respectively a hyperbola) with foci at the points $F_{1}$ and $F_{2}$.

A right circular cone is given. A plane $\Gamma$, not containing the vertex of the cone, makes with the axis of the cone an angle equal to the angle between the axis of the cone and its generators. Prove that the curve of intersection of the plane $\Gamma$ and the cone is a parabola.

A segment of constant length slides with its endpoints along two mutually perpendicular lines. A point $M$ divides it into two parts of lengths $a$ and $b$. Find the curve traced by the point $M$ as the segment moves.

Two circles with center at the origin $O$ and radii $a$ and $b$, with $a > b$, are given. A ray rotates about the point $O$, intersecting the first circle at a point $A$ and the second at a point $B$. Through the points $A$ and $B$, lines are drawn perpendicular to the axes $O x$ and $O y$, respectively. Find the curve traced by the point of intersection of these lines as the ray rotates.

An ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16} = 1$ is given. Set up its equation in a polar coordinate system, taking as the pole:

Set up the equation of the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{16} = 1$ in a (generalized) polar coordinate system, taking as the pole:

Set up the equation of the parabola $y^{2} = 2 x$ in a polar coordinate system, taking as the pole the focus of the parabola, and as the direction of the polar axis the positive direction of the axis $O x$.

What curves are given by the following equations in a generalized polar coordinate system:

Let $M_{1}$ and $M_{2}$ be the points of intersection of a conic section with a line passing through the focus $F$. Prove that $\frac{1}{M_{1} F}+\frac{1}{M_{2} F} = \frac{2}{p}$, where $p$ is the focal parameter.

Through the center of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$ two mutually perpendicular diameters are drawn. Prove that the sum of the squares of the reciprocals of their lengths equals $\frac{1}{4 a^{2}}+\frac{1}{4 b^{2}}$.

Find the ratio in which the center of an ellipse or hyperbola divides the segment of its focal axis contained between a focus and the corresponding directrix.