PROBLEMS
1. Given a quadrilateral, construct the quadrangle polar to it with respect to a given conic.
2. A point moves along a straight line. Show that its polar lines with respect to two given conics generate a point-row of the second order.
3. Given five points, draw the polar of a point with respect to the conic passing through them, without drawing the conic itself.
4. Given five lines, draw the polar of a point with respect to the conic tangent to them, without drawing the conic itself.
5. Dualize problems 3 and 4.
6. Given four points on the conic, and the tangent at one of them, draw the polar of a given point without drawing the conic. Dualize.
7. A point moves on a conic. Show that its polar line with respect to another conic describes a pencil of rays of the second order.
Suggestion. Replace the given conic by a pair of protective pencils.
8. Show that the poles of the tangents of one conic with respect to another lie on a conic.
9. The polar of a point A with respect to one conic is a, and the pole of a with respect to another conic is A'. Show that as A travels along a line, A' also travels along another line. In general, if A describes a curve of degree n, show that A' describes another curve of the same degree n. (The degree of a curve is the greatest number of points that it may have in common with any line in the plane.)