| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| 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 on rectangular hyperbolas requiring parametric differentiation, normal equation derivation, and solving a quartic equation arising from re-intersection conditions. While the techniques are standard for FM students, the multi-step algebraic manipulation and the non-trivial re-intersection condition (leading to a quartic that factors) place it well above average difficulty. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
\begin{enumerate}
\item In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
\end{enumerate}
The rectangular hyperbola $H$ has equation $x y = 20$\\
The point $P \left( 2 t \sqrt { a } , \frac { 2 \sqrt { a } } { t } \right) , t \neq 0$, where $a$ is a constant, is a general point on $H$\\
(a) State the value of $a$\\
(b) Show that the normal to $H$ at the point $P$ has equation
$$t y - t ^ { 3 } x - 2 \sqrt { 5 } \left( 1 - t ^ { 4 } \right) = 0$$
The points $A$ and $B$ lie on $H$\\
The point $A$ has parameter $t = c$ and the point $B$ has parameter $t = - \frac { 1 } { 2 c }$, where $c$ is a constant.
The normal to $H$ at $A$ meets $H$ again at $B$\\
(c) Determine the possible values of $C$
\hfill \mbox{\textit{Edexcel F1 2023 Q6 [9]}}