# The Circle of Apollonius... Coordinate Edition

##### Age 16 to 18

Challenge Level

*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 $\lambda>0$.

- Can you sketch the locus of the point $P$ for different values of $\lambda$?
- 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.

**This is an Underground Mathematics resource.**
*Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.*
*Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.*