| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| 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 differentiation (implicit or parametric), finding the normal equation, and solving a cubic equation. While it involves multiple steps and the parametric form, the techniques are routine for FP1 students and the algebraic manipulation, though somewhat involved in part (b), follows a predictable pattern. It's moderately harder than average A-level due to being Further Maths content, but straightforward within that context. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
\begin{enumerate}
\item The rectangular hyperbola $H$ has Cartesian equation $x y = 9$
\end{enumerate}
The point $P$ with coordinates $\left( 3 t , \frac { 3 } { t } \right)$, where $t \neq 0$, lies on $H$\\
(a) Use calculus to determine an equation for the normal to $H$ at the point $P$
Give your answer in the form $t y - t ^ { 3 } x = \mathrm { f } ( t )$
Given that $t = 2$\\
(b) determine the coordinates of the point where the normal meets $H$ again.
Give your answer in simplest form.
\hfill \mbox{\textit{Edexcel F1 2023 Q3 [7]}}