§ 2

Prove that the vertices of a hyperbola and the points of intersection of its asymptotes with the directrices lie on one circle.

The focus of an ellipse (hyperbola or parabola) divides a chord passing through it into segments of lengths $u$ and $v$. Prove that the sum $1 / u+1 / v$ is constant.

Derive the equations of the ellipse, the hyperbola and the parabola in a polar coordinate system, taking as the pole a focus and as the polar axis the ray lying on the axis of symmetry and not intersecting the directrix corresponding to that focus.

An ellipse and a parabola are drawn in the plane together with their axes of symmetry. How can one construct their foci and directrices with compass and straightedge? The same question for a hyperbola whose asymptotes are drawn. (The problem of constructing the axes of symmetry and the asymptotes is solved on the basis of the material of § 3.)

Let $u$ and $v$ be the lengths of two mutually perpendicular radii of an ellipse. Find the sum $1 / u^{2}+1 / v^{2}$.

Find the shortest distance from the parabola $y^{2}=12 x$ to the line $x-y+7=0$.

Prove that the segment of a tangent enclosed between the asymptotes of a hyperbola is bisected by the point of tangency.

In the equation of the tangent to an ellipse in the canonical coordinate system, instead of the coordinates $x_{0}$ and $y_{0}$ of the point of tangency there are substituted the coordinates of a point lying not on the ellipse but outside the ellipse. How is the resulting line situated?

From a point on the directrix two tangents to a parabola are drawn. Prove that they are mutually perpendicular and that the segment joining the points of tangency passes through the focus.