| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Rectangular hyperbola normal equation |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question combining rectangular hyperbola normals with parabola geometry. Part (a) is routine calculus with parametric differentiation. Part (b) requires finding the normal equation and solving simultaneously with the parabola. Part (c) uses standard parabola focus-directrix properties. While it spans multiple topics and requires careful algebra, each step follows standard Further Maths techniques without requiring novel insight. The mark allocation and structured parts make it more accessible than its length suggests. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
\begin{enumerate}
\item The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$ where $c$ is a positive constant.
\end{enumerate}
The point $P \left( c t , \frac { c } { t } \right)$, where $t > 0$, lies on $H$\\
(a) Use calculus to show that an equation of the normal to $H$ at $P$ is
$$t ^ { 3 } x - t y = c \left( t ^ { 4 } - 1 \right)$$
The parabola $C$ has equation $y ^ { 2 } = 6 x$\\
The normal to $H$ at the point with coordinates $( 8,2 )$ meets $C$ at the point $Q$ where $y > 0$\\
(b) Determine the exact coordinates of $Q$
Given that
\begin{itemize}
\item the point $R$ is the focus of $C$
\item the line $l$ is the directrix of $C$
\item the line through $Q$ and $R$ meets $l$ at the point $S$\\
(c) determine the exact length of $Q S$
\end{itemize}
\hfill \mbox{\textit{Edexcel F1 2024 Q9 [13]}}