Loci of Points; Deriving Equations of Curves in the Plane

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.

A circle of radius $a$ with center at a point $O$ and its diameter $B C$ are given. A variable point $A$ moves along the circle. On the ray $O A$ a segment $O M$ is laid off, equal to the distance from the point $A$ to the line $B C$. Find the curve traced by the point $M$.

A point $O$ and a line $\ell$ at a distance $a$ from $O$ are given. A ray rotates about the point $O$, intersecting the line $\ell$ at a variable point $P$. On this ray, a segment $O M$ is laid off from the point $O$ such that $\left|O M\right| \cdot \left|O P\right| = b^{2}$. Find the curve traced by the point $M$ as the ray rotates.

In a circle of radius $a$ a diameter $O A$ is drawn. A ray rotates about its endpoint $O$, intersecting the circle at a variable point $B$. On the extension of the chord $O B$ beyond the point $B$, a segment $B M$ equal to $A B$ is laid off. Find the curve traced by the point $M$ as the ray rotates.

Two vertices of a triangle are fixed at the points $A$ and $B$, with $\left|A B\right| = c$; the third vertex moves along a circle of radius $b$ centered at the point $A$. What curve is traced by the point $D$ where the bisector of the angle $A$ meets the side $B C$?

Let $\Gamma$ be some algebraic curve. Prove that any line either intersects $\Gamma$ in only finitely many points, or is entirely contained in $\Gamma$.

Prove that the curve $y = \sin x$ is not algebraic.

(Archimedean spiral.) A ray rotates about a point $O$ with constant angular velocity $\omega$. A point $M$ moves along it with constant speed $v$. Set up the equation, in polar coordinates, of the curve traced by the point $M$, if at the initial moment of motion the ray coincides with the positive direction of the abscissa axis, and the point $M$ with the origin $O$. Sketch the resulting curve.

Prove that in the preceding problem the Archimedean spiral is tangent to the rotating ray at the initial moment of time.

Prove that the Archimedean spiral is not an algebraic curve.

(Lemniscate of Bernoulli.) Two points $F_{1}$ and $F_{2}$ are given, at a distance $2 c$ from each other. Find the equation of the locus of points for which the product of the distances to the points $F_{1}$ and $F_{2}$ equals $c^{2}$, taking as the origin of coordinates the midpoint of the segment $F_{1} F_{2}$, and as the abscissa axis the line $F_{1} F_{2}$. Pass to the corresponding polar coordinate system. Sketch the curve.

At what angle does the lemniscate of Bernoulli from the preceding problem intersect the axis $O x$:

(Oval of Descartes.) In a polar coordinate system, find the equation of the locus of points whose distances $\rho_{A}$ and $\rho_{B}$ to two fixed points $A$ and $B$, at a distance $c$ from each other, satisfy the equality $\rho_{B} = a \rho_{A}+b$, taking $A$ as the pole, and the line $A B$ as the polar axis.

In right triangles formed by the coordinate axes and the lines intersecting them, lying in the first and third quadrants and having one and the same area $s > 0$, perpendiculars are dropped from the origin onto the hypotenuse. Find the locus of the feet of these perpendiculars. Sketch it.

(Cissoid of Diocles.) On a circle of radius $a$, two diametrically opposite points $O$ and $K$ are taken, and at the point $K$ a tangent to the circle is drawn. A ray rotates about the point $O$, intersecting the circle and the tangent line at points $A$ and $B$, respectively. On this ray, a segment $\left|O M\right| = \left|A B\right|$ is laid off from the point $O$. Write the equation of the curve traced by the point $M$ as the ray rotates, in polar and rectangular coordinates, taking the point $O$ as the origin, and the direction $\overrightarrow {O K}$ as the positive direction of the axis $O x$. Sketch the curve.

Under the conditions of the preceding problem, prove that the cissoid of Diocles is tangent to the axis $O x$, and that the line $x = 2 a$ is its asymptote.

Problem 233: on a circle of radius $a$, two diametrically opposite points $O$ and $K$ are taken, and at the point $K$ a tangent to the circle is drawn. A ray rotates about the point $O$, intersecting the circle and the tangent line at points $A$ and $B$, respectively. On this ray, a segment $\left|O M\right| = \left|A B\right|$ is laid off from the point $O$; the curve traced by the point $M$ as the ray rotates is the cissoid of Diocles, with origin at the point $O$ and positive direction of the axis $O x$ along $\overrightarrow {O K}$.

(Conchoid of Nicomedes.) A point $O$ and a line $\ell$ at a distance $a$ from the point $O$ are given. A ray rotates about the point $O$. Let the line containing this ray intersect the line $\ell$ at a variable point $B$. On the line $O B$, from the point $B$ in the direction of the ray, a segment $B M$ is laid off, $\left|B M\right| = b$. Find the equation of the curve traced by the point $M$ as the ray rotates, in generalized polar and rectangular coordinates, taking the point $O$ as the origin, and as the positive direction of the axis $O x$ the ray $\overrightarrow {O A}$, where $A$ is the foot of the perpendicular dropped from the point $O$ to the line $\ell$. Sketch this curve.

(Limaçon of Pascal.) On a circle, two diametrically opposite points $O$ and $A$ are taken, with $\left|O A\right| = a$. A ray rotates about the point $O$. Let $B$ be the variable point of intersection of the line containing this ray with the circle. On the line $O B$, from the point $B$ in the direction of the ray, a segment $B M$ of length $b$ is laid off. Write, in generalized polar and rectangular coordinates, the equation of the curve traced by the point $M$ as the ray rotates, taking the point $O$ as the origin and the direction $\overrightarrow {O A}$ as the positive direction of the axis $O x$.

Indicate the angle at which the limaçon of Pascal from the preceding problem intersects the axis $O x$ at the origin

Prove that the locus of the feet of the perpendiculars dropped from a given point onto the tangents to a given circle is a limaçon of Pascal.

(Cardioid.) A circle of radius $a$ is given, and on it two diametrically opposite points $O$ and $A$. A ray rotates about the point $O$. Let the line containing this ray intersect the circle at a variable point $B$. On this line, from the point $B$ in the direction of the ray, a segment $B M$ equal to the diameter of the circle is laid off. Write, in generalized polar and rectangular coordinates, the equation of the curve traced by the point $M$ as the ray rotates, taking the point $O$ as the origin, and the ray $O A$ as the positive direction of the axis $O x$.

(Strophoid.) A point $O$ and a line $\ell$ are given. Let $A$ be the foot of the perpendicular dropped from the point $O$ to the line $\ell$, with $\left|O A\right| = a$. A ray rotates about the point $O$, and on the line containing this ray, from the point $B$ of its intersection with the line $\ell$, segments $\left|B M_{1}\right| = \left|B M_{2}\right| = \left|A B\right|$ are laid off. Find the equation of the curve traced by the points $M_{1}$ and $M_{2}$ as the ray rotates, in generalized polar and rectangular coordinates, taking the point $O$ as the origin, and the ray $O A$ as the positive direction of the axis $O x$.

At what angle does the strophoid from the preceding problem intersect the axis $Ox$:

A segment of constant length $2 a$ slides with its endpoints along two mutually perpendicular lines. Find, in generalized polar and rectangular coordinates, the equation of the curve traced during this motion of the segment by the foot of the perpendicular dropped from the point of intersection of the lines onto the segment.

(Astroid.) A circle with center $O$ and radius $a$ is given, together with two perpendicular lines $O x$ and $O y$. A point $A$ moves along the circle, and from it perpendiculars are dropped: $A B$ onto $O x$, $A C$ onto $O y$, and $A M$ onto $B C$. Write the parametric equation of the curve traced by the point $M$ as the point $A$ moves along the circle, taking as the parameter the angle $t$ formed by the ray $O A$ with the axis $O x$. Write the equation of this curve in Cartesian coordinates. Sketch it.

Prove that the astroid is an algebraic curve and find the algebraic equation defining it, under the conditions of the preceding problem.

Problem 243: a circle with center $O$ and radius $a$ is given, together with two perpendicular lines $O x$ and $O y$; the astroid is the curve traced by the point $M$ in this coordinate system, with parametric equation $x = a \cos^{3} t, y = a \sin^{3} t$.

(Witch of Agnesi.) On a circle of diameter $a$, two diametrically opposite points $O$ and $K$ are taken, and at the point $K$ a tangent to the circle is drawn. A ray rotates about the point $O$, intersecting the circle and the tangent line at points $A$ and $B$, respectively. Through the point $A$ a line parallel to the tangent is drawn, and through the point $B$ a line parallel to the diameter $O K$. Write the parametric equation of the curve traced by the point $M$ of intersection of these lines, taking the point $O$ as the origin, the direction $\overrightarrow {O K}$ as the positive direction of the axis $O x$, and as the parameter the angle $\varphi$ formed by the ray $O A$ with the ray $O K$. Write the resulting equation in Cartesian coordinates. Sketch the curve.

(Cycloid.) A circle of radius $a$ rolls without slipping along the axis $O x$. Set up the parametric equation of the curve traced by the point $M$ of the rolling circle that at the initial moment was at the origin, taking as the parameter $t$ the angle from the radius $C M$ of the circle, going to the point $M$, to the radius $C A$, going to the point of tangency $A$.

(Epicycloid.) A circle of radius $r$ rolls without slipping along a circle of radius $R$, remaining outside it. Find the parametric equation of the curve traced by a point of the rolling circle, taking as the origin the center of the fixed circle, and as the parameter the angle $t$ between the positive direction of the abscissa axis and the radius of the fixed circle going to the point of tangency of the two circles. In the initial position, the point of tangency of the circles lies on the abscissa axis.

(Hypocycloid.) A circle of radius $r$ rolls without slipping along a circle of radius $R$, remaining inside it. Write the parametric equation of the curve traced by a point of the rolling circle. The choice of coordinate system and notation is the same as in the preceding problem.

Show that:

A circle of radius $R$ is given. Prove that:

A segment of constant length moves so that one of its endpoints slides along the circle $x^{2}+y^{2} = r^{2}$, while the other slides along the axis $O x$ (a slider-crank mechanism). Set up the equation of the curve traced by the point of the segment that divides it into parts of lengths $a$ and $b$.

(Involute of a circle.) A line rolls without slipping along the circle $x^{2}+y^{2} = r^{2}$, its initial position being $x = r$. Set up the parametric equations of the trajectory of a point of the rolling line, if at the initial moment the point has coordinates $(r, 0)$, taking as the parameter the angle from the positive direction of the axis $O x$ to the position vector of the point of tangency of the line, measured counterclockwise.

Three vertices of a rhombus lie on the sides $A B, B C$ and $C D$ of a given square, respectively. Where can the fourth vertex lie?

Prove that the following curves are cardioids:

A circle is given, and on it a point $A$. Prove that the circles passing through the point $A$, whose centers lie on the given circle, sweep out a region bounded by a certain cardioid.

The vertex $O$ of a hinged parallelogram $O P M Q$ is fixed, and the sides $O P$ and $O Q$ rotate with equal angular speed in opposite directions. Along what curve does the vertex $M$ move?

Two lines in the plane rotate in opposite directions with equal angular speed about the points $A$ and $B$ lying on them. Along what curve does their point of intersection move?

A segment $A B$ is given in the plane. Find the locus of points $M$ such that $\angle M B A = 2 \angle M A B$.

Is the curve $y = e^{x}$ algebraic?