Polar equation of an ellipse in polar axis with pole not in origin

Question

I need to find the polar equation for this ellipse, using OA (which is parallel to the x-axis) as polar semiaxis.

I'm wondering if there's a better way to do that than the way I've done it:

In the cartesian equation $\frac{x^2}{16}+\frac{(y-2)^2}{9}=1$, I replaced $x$ with $r\cos\theta$ and $y$ with $r\sin\theta$, expanded the result and solved for $r$ using the quadratic formula, which ended up working, but was a lot of work.

Answer 1

The parametric equations are

$$x=4\cos (t ) $$ $$y=2+3\sin(t) $$