FP1 January 2012 Q9
9. The rectangular hyperbola \(H\) has cartesian equation \(xy = 9\)
The points \(P\left(3p,\ \dfrac{3}{p}\right)\) and \(Q\left(3q,\ \dfrac{3}{q}\right)\) lie on \(H\), where \(p \neq \pm q\).
(a) Show that the equation of the tangent at \(P\) is \(x + p^2y = 6p\). (4)
(b) Write down the equation of the tangent at \(Q\). (1)
The tangent at the point \(P\) and the tangent at the point \(Q\) intersect at \(R\).
(c) Find, as single fractions in their simplest form, the coordinates of \(R\) in terms of \(p\) and \(q\). (4)
| Scheme | Marks |
|---|---|
| \(y = 9x^{-1} \Rightarrow \dfrac{\mathrm{d}y}{\mathrm{d}x} = -9x^{-2}\) \(xy = 9 \Rightarrow x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y = 0\) or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{\mathrm{d}y}{\mathrm{d}t}.\dfrac{\mathrm{d}t}{\mathrm{d}x} = \dfrac{-3}{p^2}.\dfrac{1}{3}\) \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = k\,x^{-2}\) Correct use of product rule. The sum of two terms, one of which is correct. their \(\dfrac{\mathrm{d}y}{\mathrm{d}t} \times \left(\dfrac{1}{\text{their }\frac{\mathrm{d}x}{\mathrm{d}t}}\right)\) | M1 |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = -9x^{-2}\) or \(x\dfrac{\mathrm{d}y}{\mathrm{d}x} + y = 0\) or \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = \dfrac{-3}{p^2}.\dfrac{1}{3}\) Correct differentiation. | A1 |
| \(y - \dfrac{3}{p} = -\dfrac{1}{p^2}(x - 3p)\) Applies \(y - \dfrac{3}{p} = (\text{their } m)(x - 3p)\) or \(y = (\text{their } m)x + c\) using \(x = 3p\) and \(y = \dfrac{3}{p}\) in an attempt to find \(c\). Their \(m\) must be a function of \(p\) and come from their dy/dx. | M1 |
| \(x + p^2y = 6p\) * Cso **given answer** | A1 |
| (4) |
Notes
Special case – if the correct gradient is quoted could score M0A0M1A1
| Scheme | Marks |
|---|---|
| \(x + q^2y = 6q\) Allow this to score here or in (c) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(6p - p^2y = 6q - q^2y\) Attempt to obtain an equation in one variable \(x\) or \(y\) | M1 |
| \(y(q^2 - p^2) = 6(q - p) \Rightarrow y = \dfrac{6(q - p)}{q^2 - p^2}\) \(x(q^2 - p^2) = 6pq(q - p) \Rightarrow x = \dfrac{6pq(q - p)}{q^2 - p^2}\) Attempt to isolate \(x\) or \(y\) – must reach \(x\) or \(y = \mathrm{f}(p, q)\) or \(\mathrm{f}(p)\) or \(\mathrm{f}(q)\) | M1 |
| \(y = \dfrac{6}{p + q}\) One correct simplified coordinate | A1 |
| \(x = \dfrac{6pq}{p + q}\) Both coordinates correct and simplified | A1 |
| (4) | |
| (9 marks) |