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