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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 $\lambda>0$.
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.