The hyperbola $\displaystyle \frac{x^2}{a^2}-\frac{y^2}{b^2} = 1 $ has an eccentricity of e. The ellipse $\displaystyle \frac{x^2}{a^2+b^2}+\frac{y^2}{b^2} = 1$ has an eccentricity of 1/e.

If the two graphs intersect at P in the first quadrant, show that the acute angle $\displaystyle \Theta $ between the the tangents to the curves at P satisfies:

$\displaystyle tan\Theta = \sqrt{2}(e + \frac{1}{e}) $

Thanks!