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

Factorisable Quadratics

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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:
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 $\lambda=3$.






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?




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 to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.