5. A curve has equation \(x ^ { 3 } - 3 x ^ { 2 } y + y ^ { 2 } + 1 = 0\).
- Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 x y - 3 x ^ { 2 } } { 2 y - 3 x ^ { 2 } }\).
- Find the equation of the normal to the curve at the point \(( 1,2 )\).
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\section*{6. In this question you must show detailed reasoning.}
A circle touches the lines \(y = \frac { 1 } { 2 } x\) and \(y = 2 x\) at \(( 6,3 )\) and \(( 3,6 )\) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{f9e0bca6-c2a3-4868-b38b-942ceabd4992-14_515_524_338_790}
Find the equation of the circle.
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