polar form of ellipse
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
Any help or guidance would be much appreciated :)
There are two ways of doing this: geometric or analytic.
Originally Posted by Stonehambey
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!