4 The equation of a curve, in polar coordinates, is
$$r = \mathrm { e } ^ { - 2 \theta } , \quad \text { for } 0 \leqslant \theta \leqslant \pi .$$
- Sketch the curve, stating the polar coordinates of the point at which \(r\) takes its greatest value.
- The pole is \(O\) and points \(P\) and \(Q\), with polar coordinates ( \(r _ { 1 } , \theta _ { 1 }\) ) and ( \(r _ { 2 } , \theta _ { 2 }\) ) respectively, lie on the curve. Given that \(\theta _ { 2 } > \theta _ { 1 }\), show that the area of the region enclosed by the curve and the lines \(O P\) and \(O Q\) can be expressed as \(k \left( r _ { 1 } ^ { 2 } - r _ { 2 } ^ { 2 } \right)\), where \(k\) is a constant to be found.