### Geometry of Equations

Many ways problem

# The circle of Apollonius... coordinate edition Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

## Problem 2

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.