10. The curve \(C\) has equation \(y = x ^ { 3 } - 5 x + \frac { 2 } { x } , x \neq 0\).
The points \(A\) and \(B\) both lie on \(C\) and have coordinates \(( 1 , - 2 )\) and \(( - 1,2 )\) respectively.
- Show that the gradient of \(C\) at \(A\) is equal to the gradient of \(C\) at \(B\).
- Show that an equation for the normal to \(C\) at \(A\) is \(4 y = x - 9\).
The normal to \(C\) at \(A\) meets the \(y\)-axis at the point \(P\). The normal to \(C\) at \(B\) meets the \(y\)-axis at the point \(Q\).
- Find the length of \(P Q\).