Answer
$\theta= \cos^{-1} (\pm\dfrac{1}{e})$
Work Step by Step
Given: $r=\dfrac{ed}{1-e \cos \theta}$
This can be re-written as:
$1-e\cos \theta=\dfrac{ed}{r}$
or, $\cos \theta=\dfrac{1}{e}(1-\dfrac{ed}{r})$
This gives: $\theta=\cos^{-1} (\dfrac{1}{e}-\dfrac{d}{r})$
The asympototes for the hyperbola are:
$\theta=\lim\limits_{r \to \infty}\cos^{-1} (\dfrac{1}{e}-\dfrac{d}{r})$
or, $\theta=\pm \cos^{-1} (\dfrac{1}{e})$
Hence,
$\theta= \cos^{-1} (\pm\dfrac{1}{e})$
