Answer
$\theta= \cos^{-1} (\pm\dfrac{1}{e})$
Work Step by Step
Re-arrange the given equations as follows:
$1-e (\cos \theta)=\dfrac{ed}{r}$
This implies that
$\cos \theta=\dfrac{1}{e}(1-\dfrac{ed}{r})$
$\implies \theta=\cos^{-1} (\dfrac{1}{e}-\dfrac{d}{r})$
Now, find the asympototes for the hyperbola.
Thus, we have $\theta=\lim\limits_{r \to \infty}\cos^{-1} (\dfrac{1}{e}-\dfrac{d}{r})$
Simplify: $\theta=\pm \cos^{-1} (\dfrac{1}{e})$
Therefore,
$\theta= \cos^{-1} (\pm\dfrac{1}{e})$
