Answer
$r=\dfrac{ed}{1+e \sin \theta}$
Work Step by Step
The eccentricity is defined as:
$\dfrac{|PF|}{|Pl|}=e$ ...(1)
Here, we have $|PF|=r$ and $|Pl|=d-r \sin \theta$
Now, the equation (1) becomes:
$\dfrac{|PF|}{|Pl|}=e \implies \dfrac{r}{d-r \sin \theta}=e$
or, $r=ed-er \sin \theta$
This gives:
$r(1+e \sin \theta)=ed$
Hence, $r=\dfrac{ed}{1+e \sin \theta}$
