- The curve \(C\) has equation
$$x = 3 \tan \left( y - \frac { \pi } { 6 } \right) \quad x \in \mathbb { R } \quad - \frac { \pi } { 3 } < y < \frac { 2 \pi } { 3 }$$
- Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { x ^ { 2 } + b }$$
where \(a\) and \(b\) are integers to be found.
The point \(P\) with \(y\) coordinate \(\frac { \pi } { 3 }\) lies on \(C\).
Given that the tangent to \(C\) at \(P\) crosses the \(x\)-axis at the point \(Q\). - find, in simplest form, the exact \(x\) coordinate of \(Q\).