| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find equation of tangent |
| Difficulty | Standard +0.8 This is a Further Maths question requiring implicit differentiation of a rectangular hyperbola, finding intersection points parametrically, deriving the tangent equation, and proving an invariant geometric property. While systematic, it demands multiple techniques and algebraic manipulation beyond standard A-level, with the 'show that' proof requiring insight into the independence from k. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.07m Tangents and normals: gradient and equations1.07s Parametric and implicit differentiation1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
6. The curve $H$ has equation
$$x y = a ^ { 2 } \quad x > 0$$
where $a$ is a positive constant.
The line with equation $y = k x$, where $k$ is a positive constant, intersects $H$ at the point $P$
\begin{enumerate}[label=(\alph*)]
\item Use calculus to determine, in terms of $a$ and $k$, an equation for the tangent to $H$ at $P$
The tangent to $H$ at $P$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$
\item Determine the coordinates of $A$ and the coordinates of $B$, giving your answers in terms of $a$ and $k$
\item Hence show that the area of triangle $A O B$, where $O$ is the origin, is independent of $k$
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2021 Q6 [8]}}