6 In this question you must show detailed reasoning.
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The diagram shows the curve with equation \(4 x y = 2 \left( x ^ { 2 } + 4 y ^ { 2 } \right) - 9 x\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 4 y - 9 } { 4 x - 16 y }\).
At the point \(P\) on the curve the tangent to the curve is parallel to the \(y\)-axis and at the point \(Q\) on the curve the tangent to the curve is parallel to the \(x\)-axis.
- Show that the distance \(P Q\) is \(k \sqrt { 5 }\), where \(k\) is a rational number to be determined.