Prove some properties of conic sections

by RoRi

May 31, 2016

Consider the Cartesian equation
$\frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2} = 1.$

Prove that this equation represents all conic sections symmetric about the origin with foci at $(c,0)$ and $(-c,0)$ .
Let $c$ be a fixed constant and let $S$ be the set of all such conics as $a^2$ takes on all positive values other than $c^2$ . Prove that every curve in the set $S$ satisfies the differential equation
$xy \left( \frac{dx}{dy} \right)^2 + (x^2 - y^2 - c^2) \frac{dy}{dx} - xy = 0.$
Prove that the set $S$ is self-orthogonal. This means that the set of all orthogonal trajectories of curve in $S$ is the set $S$ itself.