| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola normal equation derivation |
| Difficulty | Standard +0.8 This is a substantial Further Maths parabola question requiring implicit differentiation to find normals, solving simultaneous equations with parameters, and using perpendicularity conditions with parametric coordinates. Part (d) requires algebraic manipulation of the perpendicularity condition involving parametric expressions. While systematic, it demands careful algebraic handling across multiple parts and is more challenging than standard C1-C4 coordinate geometry. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations1.10g Problem solving with vectors: in geometry |
6. The parabola $C$ has Cartesian equation $y ^ { 2 } = 8 x$
The point $P \left( 2 p ^ { 2 } , 4 p \right)$ and the point $Q \left( 2 q ^ { 2 } , 4 q \right)$, where $p , q \neq 0 , p \neq q$, are points on $C$.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation of the normal to $C$ at $P$ is
$$y + p x = 2 p ^ { 3 } + 4 p$$
\item Write down an equation of the normal to $C$ at $Q$
The normal to $C$ at $P$ and the normal to $C$ at $Q$ meet at the point $N$
\item Show that $N$ has coordinates
$$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$
The line $O N$, where $O$ is the origin, is perpendicular to the line $P Q$
\item Find the value of $( p + q ) ^ { 2 } - 3 p q$
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2021 Q6 [16]}}