The problem is to prove that the following curve is an ellipse
I've looked about and have found the equation of the ellipse in polar form, but I can't find it's derivation or proof anywhere.
Anyway, I started this problem by subbing in
But I ended up with a horrible mess and couldn't spot how to get it into the form
There are two ways of doing this: geometric or analytic.
Geometrically, an ellipse is defined as the locus of a point which moves so that its distance from a fixed line (the directrix) is a constant multiple (the eccentricity, which must be less than 1) of its distance from a fixed point (the focus). In this case, you can write the equation as . This says that r (the distance of the point [r,θ] from the origin) is 2/3rds of its distance from the line x=a. So the locus is an ellipse, with focus at the origin, directrix the line x=a, and eccentricity 2/3.
To solve the problem analytically, start from the equation in the form . Square both sides: . Rearrange this as . Complete the square: . Finally, divide both sides by and you get the equation in the form . Again, this is the equation of an ellipse. It is not centred at the origin, but at the point . Its semi-axes are and .
Edit. Soroban got there first with the analytic version!