| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2024 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola normal equation derivation |
| Difficulty | Standard +0.8 This is a Further Maths question requiring implicit differentiation to find the normal equation (part a), then solving a cubic system to find the second intersection point (part b). While the calculus is standard, the parametric form and solving for the second intersection requires careful algebraic manipulation beyond typical A-level Core questions. |
| 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 The parabola $C$ has equation $y ^ { 2 } = \frac { 4 } { 3 } x$
\end{enumerate}
The point $P \left( \frac { 1 } { 3 } t ^ { 2 } , \frac { 2 } { 3 } t \right)$, where $t \neq 0$, lies on $C$.\\
(a) Use calculus to show that the normal to $C$ at $P$ has equation
$$3 t x + 3 y = t ^ { 3 } + 2 t$$
The normal to $C$ at the point where $t = 9$ meets $C$ again at the point $Q$.\\
(b) Determine the exact coordinates of $Q$.
\hfill \mbox{\textit{Edexcel F1 2024 Q7 [7]}}