The Circle of Apollonius... Coordinate Edition

This resource is from Underground Mathematics.
 

Two fixed points $A$ and $B$ lie in the plane, and the distance between them is $AB=2a$, where $a>0$.

A point $P$ moves in the plane so that the ratio of its distances from $A$ and $B$ is constant:
$$\frac{PA}{PB}=\lambda,$$
where $\lambda>0$.

1. Can you sketch the locus of the point $P$ for different values of $\lambda$?
2. Using Cartesian coordinates, work out (the equation of) the locus of $P$.

Suggestion

You may find it more straightforward to first work with specific values of $a$ and $\lambda$, say $a=2$ and $\lambda=3$.

Part 2

Now assuming that $\lambda\neq1$, find the radius and centre of the circle. What is the length of the tangent to this circle from the mid-point of $AB$? What shape is traced by the tangent as $\lambda$ varies?

Background

This circle is known as the circle of Apollonius, named after the Greek geometer Apollonius of Perga.