Problem 1

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 \(P\).

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