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 \(\lambda>0\).

Can you sketch the locus of the point \(P\) for different values of \(\lambda\)?

You might find it helpful to fix the value of \(a\), say \(a=2\).

Here are some values of \(\lambda\) you might consider:

\(\lambda=1\)

\(\lambda=3\)

\(\lambda=\frac{1}{3}\)

If you choose to draw things on a coordinate grid, it might make things simpler if you place \(A\) and \(B\) at convenient locations.

Using Cartesian coordinates, work out (the equation of) the locus of \(P\).

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

The question has not specified the coordinates of the points \(A\) or \(B\), so what options do we have?

In particular, what would be “nice” coordinates to choose for the points \(A\) and \(B\)?

If we let \(P\) be at \((x,y)\), how can we work out the distances \(PA\) and \(PB\) (in terms of \(x\) and \(y\)) once we have specified the coordinates of \(A\) and \(B\)?