Parametric form dy/dx

1. The curve \(C\) has equation

$$x = 3 \tan \left( y - \frac { \pi } { 6 } \right) \quad x \in \mathbb { R } \quad - \frac { \pi } { 3 } < y < \frac { 2 \pi } { 3 }$$

1. 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 \(Q\).
2. find, in simplest form, the exact \(x\) coordinate of \(Q\).

7. Given that $$x = 6 \sin ^ { 2 } 2 y \quad 0 < y < \frac { \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { A \sqrt { \left( B x - x ^ { 2 } \right) } }$$ where \(A\) and \(B\) are integers to be found.
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\includegraphics{figure_7} The diagram shows the curve with equation $$x = (y + 4)\ln (2y + 3).$$ The curve crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [3]
2. Find the gradient of the curve at each of the points \(A\) and \(B\), giving each answer correct to 2 decimal places. [5]