| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola tangent intersection problems |
| Difficulty | Standard +0.8 This is a multi-part Further Maths parabola question requiring implicit differentiation, focus identification, and coordinate geometry to find an intersection locus. While systematic, it demands fluency with conic properties and algebraic manipulation across 4 connected parts, placing it moderately above average difficulty. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.03g 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 $y ^ { 2 } = 20 x$
\end{enumerate}
The point $P$ on $C$ has coordinates ( $5 p ^ { 2 } , 10 p$ ) where $p$ is a non-zero constant.\\
(a) Use calculus to show that the tangent to $C$ at $P$ has equation
$$p y - x = 5 p ^ { 2 }$$
The tangent to $C$ at $P$ meets the $y$-axis at the point $A$.\\
(b) Write down the coordinates of $A$.
The point $S$ is the focus of $C$.\\
(c) Write down the coordinates of $S$.
The straight line $l _ { 1 }$ passes through $A$ and $S$.\\
The straight line $l _ { 2 }$ passes through $O$ and $P$, where $O$ is the origin.
Given that $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $B$,\\
(d) show that the coordinates of $B$ satisfy the equation
$$2 x ^ { 2 } + y ^ { 2 } = 10 x$$
\hfill \mbox{\textit{Edexcel F1 2021 Q8 [10]}}