Analytic Geometry - The Locus of a Moving Circle

Question

Let A(-α,0) and B(α,0) be two points on the x-axis. A fixed circle with centre at A and radius r lies entirely in a fixed circle with centre at B and radius R. If a moving circle is tangent to the two fixed circles, what's the equation of the locus of the centre of the moving circle?

Answer 1

Describe the locus in polar co-ordinates centred at the point A

$$r(\theta) = r + x(\theta)$$

where $x(\theta)$ is the radius of the moving circle

find $x(\theta)$ by noting that $x(\theta)= EF = EG$ and applying Cosine Law to the triangle ABE

$|AE| = r+x(\theta) \\ |AB|=2\alpha \\ |BE|= R-x(\theta)$