Conics and Loci Question (Hyperbolae and Circles)

A circle has the equation $x^2 + y^2 = r^2$. Tangents are drawn from a point $P(x_1,y_1)$ to the circle and these touch the circle at points $A$ and $B$. If the position of $P$ can vary and the locus of $P$ is a hyperbola, of eccentricity $e$, whose centre is the origin, show that the chord $AB$ touches another hyperbola, eccentricity $E$, where $\frac{1}{E^2}+\frac{1}{e^2}=1$.

I've found the equation of the chord to be $xx_1 + yy_1 = r^2$, but cannot progress much further. Any help much appreciated.

• Is $e$ independent of $r?$ If not, how related? – Narasimham Feb 18 '16 at 17:51
• Can you give any further help? I'm still very unsure as to how to progress. Usually it's all about eliminating parameters but it just ends up in a mess of algebra. – wrb98 Feb 18 '16 at 18:02
• A possible approach. Parametrize the $E$ hyperbola with one parameter. Find inversion with respect to this circle radius $r$ and eliminate the parameter by C discriminant method. Something extra is also involved further... – Narasimham Feb 18 '16 at 18:09
• Would it be possible to use an envelope method to get the hyperbola from the chord equation (involving partial derivatives)? – wrb98 Feb 18 '16 at 18:25
• As an aside, the midpoint of AB describes a lemniscate. – Lucian Feb 18 '16 at 23:35

The hyperbolae are related to their inversions with respect to circle $r$.