The Circle of Apollonius... Coordinate Edition
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.
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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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