Question

Find the polar equation of conic when focus is at pole

Answer 1

We have seen that geometrically the conic sections are related since they are all created by intersecting a plane with a right circular cone. In this section we will see how they are related algebraically.

Let F be a fixed point and l a fixed line in the plane. For any point P consider the two distances:

d(P, F) - the distance between P and F

d(P, l) - the distance between P and l:

We are interested in the ratio ; it is used to define the conic sections as follows:

A collection of points P in the plane such that e = is a fixed positive number is called a conic section. The number e is called the eccentricity of the conic. The line l is called the directrix of the conic, and the point F is called the focus of the conic.

If 0 < e < 1, then the conic is an ellipse

If e = 1, then the conic is a parabola

If e > 1, then the conic is an hyperbola

We have already seen the parabola defined in terms of a directrix and a focus. This definition shows that ellipses and hyperbolas can also be defined in terms of a directrix and focus.

If we position the point F at the pole and choose a directrix to be either a line parallel or a line perpendicular to the polar axis, then the polar equation of a conic turns out to have a fairly simple form.

Consider the point F located at the pole and the directrix, l, a vertical line with Cartesian equation x = d, d > 0. Let the point P have Cartesian coordinates (x,y) and polar coordinates (r,).

The left side of the equation d(P,F) = ed(P,l) is simply r, while the right side is e(d − rcos()). Hence,

r = ed − ercos())

Solve for r and obtain

r = ed/(1 + ecos())

Had we chosen the directrix to be the vertical line with Cartesian equation x = −d (so the directrix would be to the left of the pole), we would have found the equation of the conic to be

r = ed/(1 − ecos())

You might like to verify that this is indeed the equation. We obtain a similar equation if we take the directrix to be parallel to the polar axis. For example, if the directrix is the horizontal line with Cartesian equation y = d, d > 0, we get the equation

r = ed/(1 + esin())