| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Rectangular hyperbola normal re-intersection |
| Difficulty | Challenging +1.2 This is a standard Further Pure 1 rectangular hyperbola question requiring implicit differentiation to find the normal equation (part a is a 'show that' with clear target), then solving a cubic to find the re-intersection point. While it involves multiple steps and algebraic manipulation, the techniques are routine for FP1 students and the structure is predictable. Slightly above average difficulty due to the algebraic complexity in part (b), but not requiring novel insight. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
\begin{enumerate}
\item The rectangular hyperbola $H$ has equation $x y = 64$
\end{enumerate}
The point $P \left( 8 p , \frac { 8 } { p } \right)$, where $p \neq 0$, lies on $H$.\\
(a) Use calculus to show that the normal to $H$ at $P$ has equation
$$p ^ { 3 } x - p y = 8 \left( p ^ { 4 } - 1 \right)$$
The normal to $H$ at $P$ meets $H$ again at the point $Q$.\\
(b) Determine, in terms of $p$, the coordinates of $Q$, giving your answers in simplest form.
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\hfill \mbox{\textit{Edexcel F1 2020 Q5 [9]}}