Polarity

Find the polar of the point $A$ with respect to the second-order curve given by the equation $F(x, y) = 0$ in each of the following cases:

Prove that the directrices of a second-order curve are the polars of the corresponding foci with respect to the given curve.

Find the pole of the line $\ell$ with respect to the second-order curve $F$ in each of the following cases:

Find the equation of the line passing through the point $(2,1)$ and polar-conjugate to the line $4 x-y+30 = 0$ with respect to the second-order curve $x^{2}-6 x y+9 y^{2}-12 x+14 y-7 = 0$.

On the line $x+5 y-18 = 0$, find the point polar-conjugate to the point $(-5,4)$ with respect to the second-order curve $2 x y-6 x+4 y-1 = 0$.

Given two second-order curves: the parabola $A x^{2}-2 y = 0$ and the hyperbola $2 x y = A$. Find the point whose polar with respect to both curves would be the same line.

Let a parametrization of the form (9.2) be given on an oval Г. Considering Г as a projective line and treating the parameter $t$ as an affine coordinate on it, we define the cross ratio of four points of the oval. Prove that the quadruple of points $A, B, C, D$ is harmonic if and only if the lines $A C$ and $B D$ are polar-conjugate with respect to the given oval.

Show that if, from each point of a line perpendicular to the axis of a second-order curve, a perpendicular is dropped onto the polar of that point, then all such perpendiculars pass through one and the same point, lying on the axis of the curve.

Show that if two lines pass through the focus of a second-order curve, then, for them to be polar-conjugate, it is necessary and sufficient that they be mutually perpendicular.

Find the second-order curve for which the axis Оу is the polar of the point $A(5,0)$, and the axis $O x$ is the polar of the point $B(0,3)$, given that it passes through the points $M_{1}(1,2)$ and $M_{2}(0,3 / 2)$.

Determine the locus of poles of the tangents to the circle $(x-1)^{2}+(y-2)^{2} = 16$ with respect to the circle $x^{2} + y^{2} = 25$.

A point $M$ moves along the circle $x^{2}+y^{2} = 1$. What curve is traced by the point of intersection of the tangent to the circle at this point with the polar of the same point with respect to the hyperbola $x^{2}-y^{2} = 1$?

Show that if tangents are drawn from an external point to a non-degenerate second-order curve, then the line joining this point to the midpoint of the chord of contact is a diameter of the curve conjugate to the direction of this chord.