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 \(\left(\dfrac{3}{2}a,0\right)\).
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 \(Q\).
Show that this locus is also the locus of points which are equidistant from the line \(4x+3a=0\) and the point \(\left(\dfrac{3}{4}a,0\right)\).
