| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola normal intersection problems |
| Difficulty | Challenging +1.2 This is a standard Further Maths parabola question requiring parametric differentiation, normal equation derivation, solving a cubic for intersection points, and triangle area calculation. While it involves multiple steps and Further Maths content (making it harder than typical A-level), the techniques are routine for F1 students with no novel problem-solving required. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03e Complete the square: find centre and radius of circle1.03f Circle properties: angles, chords, tangents1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations |
4. The parabola $C$ has cartesian equation $y ^ { 2 } = 12 x$
The point $P \left( 3 p ^ { 2 } , 6 p \right)$ lies on $C$, where $p \neq 0$
\begin{enumerate}[label=(\alph*)]
\item Show that the equation of the normal to the curve $C$ at the point $P$ is
$$y + p x = 6 p + 3 p ^ { 3 }$$
This normal crosses the curve $C$ again at the point $Q$.\\
Given that $p = 2$ and that $S$ is the focus of the parabola, find
\item the coordinates of the point $Q$,
\item the area of the triangle $P Q S$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2015 Q4 [14]}}