2024 CCHY P1 Q12

The positive \(x\)- and \(y\)-axes are tangents to a circle \(C\) with equation \((x - a)^2 + (y - b)^2 = r^2\).

1. What condition(s) must apply to the constants \(a\), \(b\) and \(r\)?[1]
2. It is given that \(N(1, 2)\) is a point on the circle and it is further from the centre of the circle \(C\) as compared to the origin. Show that the coordinates of the centre \(M\) are \((5, 5)\).[4]
3. Find the equation of the tangent to \(C\) at the point \(N\).[3]
4. The tangent intersects the axes at the points \(P\) and \(Q\). The point \(R\) has coordinates \((k^2, -k)\), where \(a > 2\). Find the area of the quadrilateral \(MPQR\) in terms of \(k\).[4]