| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola tangent equation derivation |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question testing standard parabola properties (focus, directrix) and routine calculus for finding tangent equations. Part (c) explicitly asks to 'show' a given result using differentiation, making it mechanical verification rather than problem-solving. Part (d) requires solving simultaneous equations with a hyperbola, which is routine algebraic manipulation. While it's a multi-part question worth several marks, each component uses standard techniques without requiring novel insight—slightly easier than average even for Further Maths. |
| Spec | 1.03h Parametric equations: in modelling contexts1.07s Parametric and implicit differentiation |
\begin{enumerate}
\item The parabola $C$ has equation $y ^ { 2 } = 32 x$ and the point $S$ is the focus of this parabola. The point $P ( 2,8 )$ lies on $C$ and the point $T$ lies on the directrix of $C$. The line segment $P T$ is parallel to the $x$-axis.\\
(a) Write down the coordinates of $S$.\\
(b) Find the length of $P T$.\\
(c) Using calculus, show that the tangent to $C$ at the point $P$ has equation
\end{enumerate}
$$y = 2 x + 4$$
The hyperbola $H$ has equation $x y = 4$. The tangent to $C$ at $P$ meets $H$ at the points $L$ and $M$.\\
(d) Find the exact coordinates of the points $L$ and $M$, giving your answers in their simplest form.\\
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\hfill \mbox{\textit{Edexcel F1 2018 Q6 [12]}}