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\includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-18_1006_938_269_591} 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 \(\frac { 5 } { 2 } \sqrt { 5 }\).

1. Write down the equation of circle \(P\).
2. Find the equation of the tangent to the circle \(P\) at \(A\).
3. 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.
4. Find the coordinates of the points of intersection of the tangent and circle \(Q\), giving the answers in surd form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.