6 The curve \(C\) has Cartesian equation \(x ^ { 2 } + x y + y ^ { 2 } = a\), where \(a\) is a positive constant.
- Show that the polar equation of \(C\) is \(r ^ { 2 } = \frac { 2 a } { 2 + \sin 2 \theta }\).
- Sketch the part of \(C\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
The region \(R\) is enclosed by this part of \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
- It is given that \(\sin 2 \theta\) may be expressed as \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta }\). Use this result to show that the area of \(R\) is
$$\frac { 1 } { 2 } a \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \tan ^ { 2 } \theta } { 1 + \tan \theta + \tan ^ { 2 } \theta } \mathrm {~d} \theta$$
and use the substitution \(t = \tan \theta\) to find the exact value of this area.