Pencils and Bundles of Second-Order Curves

Without finding the point of intersection of the circles

set up the equation of their common chord.

Two second-order curves intersect at four points $P, Q, R, S$. It is known that the axes of the canonical coordinate system for the first curve are parallel to the canonical axes of the second. Prove that the points $P, Q, R, S$ lie on one circle.

Prove that for four points of the parabola $y = k x^{2}$ to lie on one circle, it is necessary and sufficient that the sum of the abscissas of these points equal zero.

Two ellipses are given whose equations differ only in their constant terms. Prove that they are homothetic.

It is known that the equations of two hyperbolas differ only in their constant terms. Prove that they have common asymptotes.

The equations of two parabolas differ only in their constant terms. Prove that these parabolas differ by a shift along their common axis.

A second-order curve $3 x^{2}-2 x y+y^{2}+6 x-9 = 0$ is given. Find a second-order curve whose axes of symmetry coincide with the axes of the given curve, and whose semi-axes are twice as large.

Find the hyperbola symmetric to the rectangular (equilateral) hyperbola $x^{2}-6 x y-y^{2}+12 x-6 y+5 = 0$ with respect to its asymptotes.

Which of the following conditions on a second-order curve $\Gamma$ can be written as linear homogeneous relations on the coefficients of its equation? In each case where this is so, find the number of linearly independent relations corresponding to the given condition:

Prove that if a nondegenerate second-order curve passes through the vertices of a triangle that is not right-angled, and through the point where its altitudes meet, then this curve is a rectangular (equilateral) hyperbola.

Prove that if two rectangular (equilateral) hyperbolas intersect at four points, then each of these points is the orthocenter of the triangle formed by the other three points.

Prove that every second-order curve passing through the four points of intersection of two rectangular (equilateral) hyperbolas is a rectangular hyperbola or a pair of mutually perpendicular lines.

(Nine-point circle.) Prove that the centers of the rectangular (equilateral) hyperbolas circumscribed about a given triangle, together with the feet of its altitudes, form a circle passing through the midpoints of the sides of the triangle and the midpoints of the segments joining its vertices to the orthocenter.

Find the locus of the centers of hyperbolas passing through two fixed points $P, Q$ and having the same asymptotic directions.

Write the equation of the hyperbola that is tangent to the axis $O x$ at the point $P(3,0)$, has the axis $O y$ as an asymptote, and passes through the point $Q(1,1)$.

Set up the equation of a second-order curve that is tangent to the lines $x-1 = 0$ and $x+y-2 = 0$ at the points $(1,0)$ and $(0,2)$ and passes through the point $(0,0)$.

Set up the equation of a second-order curve passing through the origin and tangent to the line $4 x+3 y+2 = 0$ at the point $O_{1}(1,-2)$, and to the line $x-y-1 = 0$ at the point $O_{2}(0,-1)$.

A triangle $M O N: O(0,0), M(8,0), N(0,6)$ is given. Write the equation of a second-order curve passing through the vertex $O$ of this triangle, intersecting the sides $O M$ and $O N$ at their midpoints, and tangent to the side $M N$ at its midpoint.

Let $N_{1}, N_{2}, N_{3}, N_{4}$ be four points, no three of which lie on one line, and let $f_{i j}(x, y) = 0$ be the equation of the line joining the points $N_{i}$ and $N_{j}$. Show that any second-order curve passing through the points $N_{1}, \ldots , N_{4}$ is given by an equation of the form $\lambda f_{12}(x, y) f_{34}(x, y)+\mu f_{14}(x, y) f_{23}(x, y) = 0$.

Show that if two second-order curves have five common points, no four of which lie on one line, then they coincide.

Set up the equation of a second-order curve passing through the points of intersection of the lines $x+y-4 = 0,16 x+31 y-48 = 0$ with the lines $y-2 = 0, y+2 = 0$ and tangent to the axis $O x$.

Write the equation of a second-order curve passing through the five points $(0,0),(0,1),(1,0),(2,-5),(-5,2)$.

Prove that if a third-order curve $f(x, y) = 0$ has seven common points with a nondegenerate second-order curve $g(x, y) = 0$, then it splits into the curve $g(x, y) = 0$ and a line.

Let $\Gamma$ be a nondegenerate second-order curve, $P_{1}, P_{2}, \ldots , P_{6}$ the vertices of a hexagon inscribed in it, and $f_{i j}(x, y) = 0$ the equation of the line $P_{i} P_{j}$. Find $\lambda$ for which the curve

splits into the curve $\Gamma$ and a line.

Let $f_{i j}(x, y) = 0$ be the equations of the lines from the preceding problem. Prove Pascal's theorem: the three points of intersection of the lines

lie on one line.

Set up the equation of a parabola tangent to all sides of the quadrilateral with vertices $M_{1}(0,0), M_{2}(6,0), M_{3}(5,3), M_{4}(0,2)$.

Determine the locus of the centers of second-order curves tangent to the sides of a quadrilateral $N_{1} N_{2} N_{3} N_{4}$, if the coordinates of the vertices $N_{1}(4,0), N_{2}(0,2), N_{3}(-3,0), N_{4}(0,-1)$ are given.