## Parabolas of Apollonius?

This question is inspired by Maria Monks’ blog post on the Circles of Apollonius.

Pictured above are some parabolas from two families of parabolas.

The Blue Family is given by the parabolas $x = a - \frac{1}{4a}y^2$, for $a > 0$.

The Red Family is given by the same equation, $x = a - \frac{1}{4a}y^2$, but now $a < 0$.

Problem: Pick one Red and one Blue parabola. Show that they intersect at right angles.

What’s the spiffiest proof you can give?