Challenging +1.2 This is a standard FP1 rectangular hyperbola question requiring implicit differentiation to find tangent equations, then solving simultaneous equations to find their intersection. While it involves multiple steps (finding two tangents, solving for R), the techniques are routine for Further Maths students and the parametric form guides the algebra. The 8 marks reflect length rather than conceptual difficulty, making it moderately above average but not requiring novel insight.
The point \(P \left( 2p, \frac{2}{p} \right)\) and the point \(Q \left( 2q, \frac{2}{q} \right)\), where \(p \neq -q\), lie on the rectangular hyperbola with equation \(xy = 4\).
The tangents to the curve at the points \(P\) and \(Q\) meet at the point \(R\).
Show that at the point \(R\),
$$x = \frac{4pq}{p + q} \text{ and } y = \frac{4}{p + q}.$$
[8]
The point $P \left( 2p, \frac{2}{p} \right)$ and the point $Q \left( 2q, \frac{2}{q} \right)$, where $p \neq -q$, lie on the rectangular hyperbola with equation $xy = 4$.
The tangents to the curve at the points $P$ and $Q$ meet at the point $R$.
Show that at the point $R$,
$$x = \frac{4pq}{p + q} \text{ and } y = \frac{4}{p + q}.$$
[8]
\hfill \mbox{\textit{Edexcel FP1 Q7 [8]}}