| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Rectangular hyperbola normal equation |
| Difficulty | Challenging +1.2 This is a Further Maths question on rectangular hyperbolas using parametric forms. Part (a) requires finding the equation of a chord using two parametric points (straightforward algebra). Part (b) involves showing perpendicularity conditions lead to the normal being parallel to PQ, requiring manipulation of gradients and the relationship pq = -1/r². While it requires multiple steps and some algebraic insight, the techniques are standard for FM students and the question provides significant scaffolding through its structure. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.03b Straight lines: parallel and perpendicular relationships1.07m Tangents and normals: gradient and equations |
6. The rectangular hyperbola, $H$, has cartesian equation
$$x y = 36$$
The three points $P \left( 6 p , \frac { 6 } { p } \right) , Q \left( 6 q , \frac { 6 } { q } \right)$ and $R \left( 6 r , \frac { 6 } { r } \right)$, where $p , q$ and $r$ are distinct, non-zero values, lie on the hyperbola $H$.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation of the line $P Q$ is
$$p q y + x = 6 ( p + q )$$
Given that $P R$ is perpendicular to $Q R$,
\item show that the normal to the curve $H$ at the point $R$ is parallel to the line $P Q$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2015 Q6 [10]}}