Polar equation for an ellipse that is not centred at the origin

Question

Wikipedia says the polar form of an ellipse centred at the origin is

What if the ellipse is not centred at the origin? Like its centred at (3, 4)?

Answer 1

If one of the foci is at the origin, then the polar equation for the ellipse is $$r(\theta) = \frac{\ell}{1 + \epsilon \cos (\theta - \theta_0)},$$ where $\ell$ is the semi-latus rectum (for the ellipse you give, this is $\frac{b^2}{a}$), $\epsilon$ is the eccentricity, and $\theta_0$ is a reference angle (for the ellipse you give, this is $0$). (In fact this formula works for other conic sections just as well.)

If the origin is inside or on the ellipse but not at a focus, the formula is generally unpleasantly complicated.

If the origin is outside the ellipse, then the ellipse is not the graph of any (continuous) polar function (with period $2 \pi$), as some rays from the origin intersect the ellipse more than once.