Following on from problem 1, we have two fixed points \(A\) and \(B\) in the plane with \(AB=2a\), where \(a>0\).

A point \(P\) moves in the plane so that \[\frac{PA}{PB}=\lambda,\] where \(\lambda>0\).

We shall now assume that \(\lambda\ne1\), so that the locus of \(P\) is a circle.

What is the length of the tangent to this circle from the mid-point of \(AB\), that is, the length \(MN\) in the applet below?

What is the locus of \(N\) as \(\lambda\) varies?

*If you have not already done so, you will first need to find the radius and centre of the circle traced by \(P\).*

(In this applet, \(\lambda\) may be adjusted and the point \(N\) will be traced if you click the button. \(C\) is the centre of the circle that \(P\) describes.)

The locus of \(P\) is known as the *circle of Apollonius*. See the Historical background for this problem to read more about Apollonius and this circle.