- The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.
The point \(P \left( c t , \frac { c } { t } \right)\), where \(t > 0\), lies on \(H\)
- Use calculus to show that an equation of the normal to \(H\) at \(P\) is
$$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$
The parabola \(C\) has equation \(y ^ { 2 } = 6 x\)
The normal to \(H\) at the point with coordinates \(( 8,2 )\) meets \(C\) at the point \(Q\) where \(y > 0\) - Determine the exact coordinates of \(Q\)
Given that
- the point \(R\) is the focus of \(C\)
- the line \(l\) is the directrix of \(C\)
- the line through \(Q\) and \(R\) meets \(l\) at the point \(S\)
- determine the exact length of \(Q S\)