| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2023 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola area calculations |
| Difficulty | Challenging +1.8 This is a substantial Further Maths parabola question requiring knowledge of focus-directrix properties, coordinate geometry, and algebraic manipulation across three connected parts. Part (a) uses the focus-directrix definition, part (b) requires finding line equations and intersections, and part (c) involves area calculations leading to solving a quartic equation. While the techniques are standard for FM students, the multi-step nature, algebraic complexity, and need to synthesize several concepts makes this significantly harder than average A-level questions but not exceptionally difficult for Further Maths. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.03g Parametric equations: of curves and conversion to cartesian |
\begin{enumerate}
\item A parabola $C$ has equation $y ^ { 2 } = 4 a x$ where $a$ is a positive constant.
\end{enumerate}
The point $S$ is the focus of $C$\\
The line $l _ { 1 }$ with equation $y = k$ where $k$ is a positive constant, intersects $C$ at the point $P$\\
(a) Show that
$$P S = \frac { k ^ { 2 } + 4 a ^ { 2 } } { 4 a }$$
The line $l _ { 2 }$ passes through $P$ and intersects the directrix of $C$ on the $x$-axis.\\
The line $l _ { 2 }$ intersects the $y$-axis at the point $A$\\
(b) Show that the $y$ coordinate of $A$ is $\frac { 4 a ^ { 2 } k } { k ^ { 2 } + 4 a ^ { 2 } }$
The line $l _ { 1 }$ intersects the directrix of $C$ at the point $B$\\
Given that the areas of triangles $B P A$ and $O S P$, where $O$ is the origin, satisfy the ratio
$$\text { area } B P A \text { : area } O S P = 4 k ^ { 2 } : 1$$
(c) determine the exact value of $a$
\hfill \mbox{\textit{Edexcel F1 2023 Q8 [11]}}