Review question

# What's the locus of $(p, q)$ if the length of this tangent equals $p$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9768

## Suggestion

The fixed points $A$ and $B$ have coordinates $(-3a,0)$ and $(a,0)$ respectively. Find the equation of the locus of a point $P$ which moves in the coordinate plane so that $AP=3PB$. Show that the locus is a circle, $S$, which touches the axis of $y$ and has its centre at the point $\left(\dfrac{3}{2}a,0\right)$.

If we call $P=(x,y)$, can we write down expressions for the distances $AP$ and $PB$?

A point $Q$ moves in such a way that the perpendicular distance of $Q$ from the axis of $y$ is equal to the length of a tangent from $Q$ to the circle $S$. Find the equation of the locus of $Q$.

We could get bogged down in some very heavy algebra here if we pick the wrong method. Could we look at this problem geometrically?

Can we see a pentagon containing three right angles? What are the sides?