Answer
$r= \dfrac{4}{1 - 2 \cos (θ)}$
Work Step by Step
Convert the polar equation into Cartesian form:
The directrix is $r = -2sec(θ)$
or, $r cos(θ) = x$
Now, multiply both sides by $\cos θ$.
we have $r \cosθ = -2 \implies x = -2$
since the directrix is a vertical line, we have $r= \dfrac{ed}{1 - e \cos (θ)}$
and $ x =-2$
Given: $e = 2, d=2$
Now, we have $r= \dfrac{ed}{1 - e \cos (θ)}= \dfrac{(2) (2)}{1 - (2) \cos θ}$
Hence, $r= \dfrac{4}{1 - 2 \cos θ}$
