| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola normal equation derivation |
| Difficulty | Standard +0.3 This is a standard FP1 conic sections question requiring implicit differentiation to find the normal equation (routine calculus), then solving simultaneous linear equations to find intersection point R. While it involves parametric coordinates and algebraic manipulation, these are well-practiced techniques at this level with no novel insight required. Slightly easier than average due to being a textbook-style multi-part question with clear structure. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations |
The parabola $C$ has equation $y^2 = 4ax$, where $a$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation for the normal to $C$ at the point $P(ap^2, 2ap)$ is $y + px = 2ap + ap^3$. [4]
\end{enumerate}
The normals to $C$ at the points $P(ap^2, 2ap)$ and $Q(aq^2, 2aq)$, $p \neq q$, meet at the point $R$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find, in terms of $a$, $p$ and $q$, the coordinates of $R$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q30 [9]}}