Can you work out the equation of an ellipse in polar co-ordinates?

It is easy enough to produce an ellipse in cartesian co-ordinates, by squashing a circle in the y direction.

Instead of x²/r² + y²/r² = 1
which is the normal equation of a circle, choose different values for the "radius" in each direction, to get

x²/a² + y²/b² = 1

But how do you know this is an ellipse? And where is the focus in this formulation?

The ellipse is fundamental in astronomy, where it is the shape of the orbit of a gravitationally bound object. Two objects each orbit in ellipses around the mutual centre of gravity, with the centre of gravity at one of the foci of the ellipse.

One construction for an ellipse is to put two pins into a piece of paper, which will form the foci of the ellipse, and then attach a loose piece of string to each pin. Stretching it with the pencil point, you trace the pencil point around to make the shape of the ellipse.

We can do this algebraically with the following diagram. If the length of the string is k and the distance between the pins is c (k > c) we have this arrangement:

r and x are just the polar co-ordinates of the point C about the point O.

Using just the cosine rule, we have
(k-r)² = r² + c² - 2cr cos x
k²-2kr = c² - 2cr cos x
(k² - c²)/2 = r(k - c cos x)

Now making the substitutions a=k/2 and e=c/k we get
r = a (1 - e²)/(1 - e cos x)

This is the polar equation of an ellipse from one focus, where a is the semi-major axis (half of the longest diameter) and e the eccentricity.