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# The Circle of Apollonius... Coordinate Edition

#### Suggestion

#### Part 2

#### Background

**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.*
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Age 16 to 18

Challenge Level

- Problem
- Student Solutions

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 $P$.

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

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?

This circle is known as the *circle of Apollonius*, named after the Greek geometer Apollonius of Perga.

This comes in two parts, with the first being less fiendish than the second. Itâ€™s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.