Geometry of Equations

Many ways problem

Suggestion 1

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 $\lambda>0$.

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

You might find it helpful to fix the value of $a$, say $a=2$.

Here are some values of $\lambda$ you might consider:

• $\lambda=1$

• $\lambda=3$

• $\lambda=\frac{1}{3}$

If you choose to draw things on a coordinate grid, it might make things simpler if you place $A$ and $B$ at convenient locations.

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 $\lambda=3$.

The question has not specified the coordinates of the points $A$ or $B$, so what options do we have?

In particular, what would be “nice” coordinates to choose for the points $A$ and $B$?

If we let $P$ be at $(x,y)$, how can we work out the distances $PA$ and $PB$ (in terms of $x$ and $y$) once we have specified the coordinates of $A$ and $B$?