| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola tangent intersection problems |
| Difficulty | Standard +0.3 This is a structured multi-part question on parabola tangents that guides students through standard techniques: implicit differentiation to find the tangent equation (part a is a 'show that'), identifying the directrix, substituting to find constants, and back-substituting. While it requires knowledge of parabola properties and careful algebra, each step follows directly from the previous one with no novel insight required. Slightly easier than average due to the scaffolding. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.07m Tangents and normals: gradient and equations |
8. The parabola $C$ has cartesian equation $y ^ { 2 } = 36 x$. The point $P \left( 9 p ^ { 2 } , 18 p \right)$, where $p$ is a positive constant, lies on $C$.
\begin{enumerate}[label=(\alph*)]
\item Using calculus, show that an equation of the tangent to $C$ at $P$ is
$$p y - x = 9 p ^ { 2 }$$
This tangent cuts the directrix of $C$ at the point $A ( - a , 6 )$, where $a$ is a constant.
\item Write down the value of $a$.
\item Find the exact value of $p$.
\item Hence find the exact coordinates of the point $P$, giving each coordinate as a simplified surd.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2017 Q8 [11]}}