The tangent at $\displaystyle P(a\sec\theta,b\tan\theta)$ on the hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ has equation $\displaystyle \frac{x\sec\theta}{a}-\frac{y\tan\theta}{b}=1$.

If teh tangent at P is also tangent to the circle center $\displaystyle (ae,0)$ and radius $\displaystyle ae\sqrt{e^2+1}$, show that $\displaystyle \sec\theta=-e$ where e is the eccentricity of the hyperbola.