The curve \(C\) is defined by the parametric equations \[x=a\csc\theta,\qquad y=2a\cot\theta,\qquad \frac{\pi}{2}<\theta<\pi,\] where \(a\) is a positive constant.
The line \(D\) has equation \(y=-x\) and intersects curve \(C\) once.
Find, in terms of \(a\), the exact \(x\)-coordinate of the point of intersection.[2]
Show that gradient of \(C\), \(\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}=\frac{2\csc\theta}{\cot\theta}\).[2]
The normal to \(C\) at the point \(P(a\csc p,2a\cot p)\) cuts the \(x\)-axis at point \(A\) and the \(y\)-axis at point \(B\). The point \(M\) is the midpoint of \(AB\). Find a cartesian equation of the curve traced by \(M\) as \(p\) varies.[5]
Given that the \(p\) is increasing at the rate of 0.1 unit per second, find the rate at which the area of triangle \(AOB\) is changing when \(p=2\) and \(a=1\).[3]