Miscellaneous Problems

Prove that the area contained between a tangent to a hyperbola and its asymptotes is a constant quantity.

Prove that the product of the segments of a secant between the asymptotes and a point on the hyperbola equals the square of the semi-diameter parallel to the secant.

Prove that for a hyperbola, the point of tangency bisects the segment of the tangent between the asymptotes.

Find the equation of the second-order curve from its focus $(0,0)$ and three tangents: $x-y-1=0$, $2x-y-1=0$, $x+y-1=0$.

Find the parabola from a point on it $(5,0)$, focus $(3,2)$, and tangent $x-3y-7=0$.

Find the hyperbola from its center $(1,1)$, focus $(3,3)$, and tangent $x+2y-7=0$.

Find the locus of the points of tangency of the tangents drawn from the origin to the parabola

where $m$ is a variable parameter.

Find the locus of the projections of the focus onto the tangents to the curve $\frac{x^{2}}{a^{2}}\pm \frac{y^{2}}{b^{2}}=1$.

Find the point of intersection of the polars of points lying on the directrix.

If $a$ and $b$ are the lengths of two mutually perpendicular tangents to the parabola $y^{2}=4x$, then $a^{4}b^{4}=(a^{2}+b^{2})^{3}$. Prove this.

Find the locus of the points of tangency, for tangents of a given direction, to the confocal curves $\frac{x^{2}}{a^{2}+h}+\frac{y^{2}}{b^{2}+h}=1$, where $h$ is a variable parameter.

An equilateral hyperbola passes through the origin and has the asymptote $x+y+1=0$. Find the locus of the points of intersection of the second asymptote with the tangent at the origin.

Equilateral hyperbolas pass through the point $(1,0)$ and are tangent to the axis $Oy$ at the point $(0,1)$. Find the locus of their centers.

What is the locus of the vertices of equilateral hyperbolas passing through a given point and having a given asymptote?

What is the locus of points from which two normals at right angles to one another can be drawn to a given parabola?

The parabolas $y^{2}=2px$ and $y^{2}=2p(x-a)$ are given. Prove that the chords of the first that are tangent to the second are bisected by the point of tangency.

Two parabolas have a common axis of symmetry. Two lines are parallel to the axis of symmetry. Chords are drawn through the points of their intersection with the parabolas. Prove that the points of intersection of these chords lie on one line, whichever pair of parallel lines is taken.

What is the locus of the vertices of parabolas with a given focus and tangent?

Prove that the circle circumscribed about the triangle formed by three tangents to a parabola passes through the focus.

Prove that the equilateral hyperbola circumscribed about a triangle passes through the point of intersection of its altitudes.

What is the locus of the foci of parabolas with a given vertex and tangent?

From a point outside a parabola, three normals can be drawn to it. Prove that the vertex of the parabola and the three feet of the normals lie on one circle.

The curve $Ax^{2}+By^{2}+C=0$ and the point $M(a,b)$ are given. About $M$ as center, circles are described. Find the locus of the midpoints of the chords common to these circles and the given curve.

A quadrilateral $ABCD$ is given. What is the locus of points $M$ such that the sum of the areas of the triangles $AMB$ and $CMD$ equals the sum of the areas of the triangles $BMC$ and $DMA$?

Using the result of the preceding problem, prove Newton's theorem: in a quadrilateral circumscribed about a circle, the midpoints of the diagonals and the center of this circle lie on one line.

A right angle, whose vertex $M$ lies on a second-order curve, rotates about the vertex, and $A$ and $B$ are the points of intersection of its sides with the curve, other than $M$. Prove that the line $AB$ rotates about a point lying on the normal to the curve at the point $M$.

A right angle rotates about a point $A$ inside a given circle. At the points of intersection of its sides with the circle, tangents to the circle are drawn. Find the locus of the points of intersection of these tangents.

A variable circle is tangent to an ellipse at a given point. What is the locus of the points of intersection of the common tangents?

The foci of an ellipse are joined by radius vectors to a variable point of the ellipse. What is the locus of the centers of the circles inscribed in the triangles formed by the focal chord and the radius vectors?

Inversion, or transformation by reciprocal radius vectors, consists in replacing a variable point $M$ by another point $M_{1}$, lying on the line $OM$, such that $OM \cdot OM_{1} = a^{2}$. Here $O$ is a given point, and $a$ is a given quantity. Prove that under inversion the line $A(x^{2}+y^{2})+Bx+Cy+D=0$ passes into the line $A_{1}(x^{2}+y^{2})+B_{1}x+C_{1}y+D_{1}=0$, i.e., circles or lines pass into circles or lines (not necessarily respectively).

Prove that by inversion with respect to the origin, the hyperbola $x^{2}-y^{2}=a^{2}$ is carried into the lemniscate of Bernoulli:

Prove that the inversion of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with respect to the origin, with radius of inversion equal to $\sqrt{ab}$, gives the curve $(x^{2}+y^{2})^{2}=b^{2}x^{2}+a^{2}y^{2}$.

A hyperbola congruent to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, where $a>b$, is inscribed in the angle between the coordinate axes and slides along them. Prove that its center moves along the circle $x^{2}+y^{2}=a^{2}-b^{2}$.

An ellipse with semi-axes $a$ and $b$ is tangent to the coordinate axes and slides along them. Prove that its center moves along the circle $x^{2}+y^{2}=a^{2}+b^{2}$.