| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola normal intersection problems |
| Difficulty | Challenging +1.2 This is a standard Further Pure parabola question requiring parametric coordinate manipulation and normal equations. Part (a) is routine algebra showing the chord equation. Part (b) involves finding normal equations and their intersection—more algebraic work but follows a well-established method taught in FP1. The 10 total marks reflect extended calculation rather than conceptual difficulty or novel insight. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations |
The points $P(ap^2, 2ap)$ and $Q(aq^2, 2aq)$, $p \neq q$, lie on the parabola $C$ with equation $y^2 = 4ax$, where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation for the chord $PQ$ is $(p + q) y = 2(x + apq)$ . [3]
\end{enumerate}
The normals to $C$ at $P$ and $Q$ meet at the point $R$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the coordinates of $R$ are $(a(p^2 + q^2 + pq + 2), -apq(p + q) )$. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q39 [10]}}