6. The quadratic equation
$$A x ^ { 2 } + 5 x - 12 = 0$$
where \(A\) is a constant, has roots \(\alpha\) and \(\beta\)
- Write down an expression in terms of \(A\) for
- \(\alpha + \beta\)
- \(\alpha \beta\)
The equation
$$4 x ^ { 2 } - 5 x + B = 0$$
where \(B\) is a constant, has roots \(\alpha - \frac { 3 } { \beta }\) and \(\beta - \frac { 3 } { \alpha }\)
- Determine the value of \(A\)
- Determine the value of \(B\)
The rectangular hyperbola \(H\) has equation \(x y = 36\)
The point \(P ( 4,9 )\) lies on \(H\)
- Show, using calculus, that the normal to \(H\) at \(P\) has equation
$$4 x - 9 y + 65 = 0$$
The normal to \(H\) at \(P\) crosses \(H\) again at the point \(Q\)
- Determine an equation for the tangent to \(H\) at \(Q\), giving your answer in the form \(y = m x + c\) where \(m\) and \(c\) are rational constants.
\section*{7. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
7 "}