| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Rectangular hyperbola tangent intersection |
| Difficulty | Standard +0.8 This is a Further Maths question on rectangular hyperbolas requiring parametric differentiation to derive a tangent equation (part a), then solving a system where two parametric tangents meet at a given point (part b). While the techniques are standard for FM students, the parametric approach and algebraic manipulation with two unknowns elevates this above routine calculus problems. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07m Tangents and normals: gradient and equations |
\begin{enumerate}
\item A rectangular hyperbola $H$ has equation $x y = 25$
\end{enumerate}
The point $P \left( 5 t , \frac { 5 } { t } \right) , t \neq 0$, is a general point on $H$.\\
(a) Show that the equation of the tangent to $H$ at $P$ is $t ^ { 2 } y + x = 10 t$
The distinct points $Q$ and $R$ lie on $H$. The tangent to $H$ at the point $Q$ and the tangent to $H$ at the point $R$ meet at the point $( 15 , - 5 )$.\\
(b) Find the coordinates of the points $Q$ and $R$.
\hfill \mbox{\textit{Edexcel F1 2021 Q4 [8]}}