2026 RVHS P1 Q7

Let \(a\) and \(b\) be fixed positive real numbers. A variable line \(C\) passing through \((a,b)\) with varying gradient \(m\) has equation \(y-b=m(x-a)\), where \(m<0\). The line \(C\) cuts the \(x\)-axis and \(y\)-axis at the points \(P(p,0)\) and \(Q(0,q)\) respectively.

1. 1. Show that \(A\), the area of triangle \(OPQ\), where \(O\) is the origin, is given by \[A=\frac12\left(2ab-a^2m-\frac{b^2}{m}\right)\text{ units}^2.\][2]
   2. Using differentiation, find the value of \(m\) that gives the minimum \(A\).[3]
2. As \(m\) varies, the line \(C\) rotates about point \((a,b)\) causing the points \(P\) and \(Q\) to move along the \(x\)- and \(y\)- axes, respectively. At the instant when \(m=-1\), point \(P\) is moving to the right, where the rate of change of \(p\) is \(1\) unit per second and point \(Q\) is moving downwards. At this instant, find the rate of change of \(q\), in units per second, leaving your answer in terms of \(a\) and \(b\).[4]