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?
