| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Rectangular hyperbola normal re-intersection |
| Difficulty | Challenging +1.8 This is a Further Maths question requiring implicit differentiation to find the normal equation (part a is a 'show that' requiring careful algebraic manipulation), then solving a cubic equation when the normal intersects the hyperbola again. The parametric form and algebraic complexity elevate this above standard calculus questions, but the techniques are systematic once the approach is identified. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations |
6. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where $c$ is a non-zero constant.
The point $P \left( c p , \frac { c } { p } \right)$, where $p \neq 0$, lies on $H$.
\begin{enumerate}[label=(\alph*)]
\item Show that the normal to $H$ at $P$ has equation
$$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$
The normal to $H$ at $P$ meets $H$ again at the point $Q$.
\item Find, in terms of $c$ and $p$, the coordinates of $Q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2016 Q6 [9]}}