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