Answer
$r= \frac{4}{1 - 2 cos (θ)}$
Work Step by Step
The directrix is r = -2sec(θ), so we need to convert this polar equation into cartesian form
r cos(θ) = x, so multiply both sides by cos (θ)
r cos(θ) = -2
x = -2
$r= \frac{ed}{1 - e cos (θ)}$ (cosine since directrix is vertical line, but negative since x =-2)
e = 2, d=2
$r= \frac{ed}{1 - e cos (θ)}$
$r= \frac{(2) (2)}{1 - (2) cos (θ)}$
$r= \frac{4}{1 - 2 cos (θ)}$
