Problem

The diagram shows a circle $P$ with centre $(0,2)$ and radius $10$ and the tangent to the circle at the point $A$ with coordinates $(6,10)$. It also shows a second circle $Q$ with centre at the point where this tangent meets the $y$-axis and with radius $\frac{5}{2}\sqrt{5}$.

(a) Write down the equation of circle $P$.

(b) Find the equation of the tangent to the circle $P$ at $A$.

(c) Find the equation of circle $Q$ and hence verify that the $y$-coordinates of both of the points of intersection of the two circles are $11$.

(d) Find the coordinates of the points of intersection of the tangent and circle $Q$, giving the answers in surd form.