Consider a conic section with a horizontal directrix which is at a distance above a focus at the origin. Prove that the points on satisfy the polar equation

if the is a ellipse or parabola and if is a hyperbola then the points on the right branch satisfy the polar equation

*Proof.* From Theorem 13.17 (page 501 of Apostol) (taking at the origin) we have

If we express in polar coordinates, and take to be horizontal, then we have . So, , and . Therefore,

If lies below the directrix , then , so . Therefore,

If lies above the directrix , then , so and we have

In this last case we also have since so this is a hyperbola