2020 SAJC Promo Q9

A curve \(C\) has parametric equations \(x=\ln (\sec \theta )\),\(y=\ln (\sin \theta )\), \(0 < \theta < \frac{\pi }{2}\).

1. Sketch the curve \(C\), stating the equation of any asymptotes.[2]
2. Find the equation of the tangent to the curve at the point where \(\theta =\frac{\pi }{4}\), leaving your answer in the form of \(y=ax+b\) where \(a\) and \(b\) are exact values.[5]
3. \(P\) is a point on the curve \(C\) which moves along \(C\) at a rate of \(\frac{\mathrm{d}\theta }{\mathrm{d}t}=2\) units/s. The point \(B\) has coordinates \((1,0)\) and \(OBP\)forms a triangle, where \(O\) is the origin. Find the rate of change of the area of triangle \(OBP\) when \(\theta =\frac{\pi }{3}\).[4]