| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Rectangular hyperbola normal equation |
| Difficulty | Standard +0.8 Part (a) is routine calculus requiring implicit differentiation and normal line equation (standard F1 content). Part (b) requires solving a cubic equation from xy=36 and 4x-9y+65=0, finding Q's coordinates, then computing another tangent. The multi-step nature, algebraic manipulation of the cubic, and coordination of techniques across parts elevates this above average difficulty, though it remains a standard Further Pure question without requiring novel insight. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation |
In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
The rectangular hyperbola $H$ has equation $xy = 36$
The point $P(4, 9)$ lies on $H$
\begin{enumerate}[label=(\alph*)]
\item Show, using calculus, that the normal to $H$ at $P$ has equation
$$4x - 9y + 65 = 0$$
[4]
\end{enumerate}
The normal to $H$ at $P$ crosses $H$ again at the point $Q$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine an equation for the tangent to $H$ at $Q$, giving your answer in the form $y = mx + c$ where $m$ and $c$ are rational constants.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2022 Q7 [9]}}