| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola normal intersection problems |
| Difficulty | Challenging +1.3 This is a structured Further Maths parabola question with clear scaffolding. Part (a) is routine calculus (finding normal equation using parametric form), part (b) requires solving a cubic for parameter values, and part (c) combines focus/directrix properties with coordinate geometry. While it requires multiple techniques and is longer than typical A-level questions, the scaffolding guides students through each step without requiring significant novel insight or problem-solving creativity. |
| Spec | 1.03f Circle properties: angles, chords, tangents1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
\begin{enumerate}
\item The parabola $C$ has equation $y ^ { 2 } = 36 x$
\end{enumerate}
The point $P \left( 9 t ^ { 2 } , 18 t \right)$, where $t \neq 0$, lies on $C$\\
(a) Use calculus to show that the normal to $C$ at $P$ has equation
$$y + t x = 9 t ^ { 3 } + 18 t$$
(b) Hence find the equations of the two normals to $C$ which pass through the point (54, 0), giving your answers in the form $y = p x + q$ where $p$ and $q$ are constants to be determined.
Given that
\begin{itemize}
\item the normals found in part (b) intersect the directrix of $C$ at the points $A$ and $B$
\item the point $F$ is the focus of $C$\\
(c) determine the area of triangle $A F B$
\end{itemize}
\hfill \mbox{\textit{Edexcel F1 2022 Q6 [11]}}