Find a polar equation for a comet moving in a parabolic orbit

Consider a comet moving in a parabolic orbit with focus the sun. When the comet is 100 million miles from the sun, the vector between the comet and the focus is at an angle of $\frac{\pi}{3}$ with the unit vector $N$ from the focus perpendicular to the directrix. The focus is in the negative half-plane determined by $N$ .

Find a polar equation for the orbit of the comet. Consider the sun to be at the origin. Further, compute the minimum distance from the comet to the sun.
Do part (a) in the case that the focus is in the positive half-plane determined by $N$ .

Since the orbit is parabolic, we know the eccentricity is $e = 1$ . Taking the focus to be the origin we then have the polar equation
$\begin{align*} \lVert X \rVert &= e d(X,L) \\ &= | X \cdot N - d |\\ &= d - X \cdot N \\ &= d - r \cos \frac{\pi}{3} \\ &= d - \frac{3}{2} r. \end{align*}$

Therefore,

$r = \frac{\frac{3}{2} 10^8}{1 + \cos \theta}$

and the minimum distance is

$d = \frac{7.5 \cdot 10^7}.$
For the focus in the positive half-plane determined by $N$ we have
$\lVert X \rVert = | X \cdot N + d | = r \cos \theta + d.$

Therefore,

$r = \frac{d}{1-\cos \theta} = \frac{ \frac{1}{2} 10^8}{1 - \cos \theta} = \frac{5 \cdot 10^7}{1 - \cos \theta}.$