FP1 June 2017 Q3
3. The rectangular hyperbola \(H\) has parametric equations \[x = 4t, \quad y = \frac{4}{t} \qquad t \neq 0\]
The points \(P\) and \(Q\) on this hyperbola have parameters \(t = \dfrac{1}{4}\) and \(t = 2\) respectively.
The line \(l\) passes through the origin \(O\) and is perpendicular to the line \(PQ\).
Give your answers in their simplest form. (3)
| Scheme | Marks |
|---|---|
| \(x = 4t,\ y = \dfrac{4}{t},\ t \neq 0\) | |
| \(t = \dfrac{1}{4} \Rightarrow P(1, 16), \quad t = 2 \Rightarrow Q(8, 2)\) | B1 |
| \(m(PQ) = \dfrac{2 - 16}{8 - 1}\ \{= -2\}\) | M1 |
| \(m(l) = \dfrac{1}{2}\) | |
| So, \(l: y = \dfrac{1}{2}x\) or \(2y = x\) | A1 oe |
| (3) |
Notes
B1: Coordinates for either \(P\) or \(Q\) are correctly stated. (Can be implied).
M1: Finds the gradient of the chord \(PQ\) with \(\dfrac{y_2 - y_1}{x_2 - x_1}\) then uses in \(y = -\dfrac{1}{m}x\). Condone incorrect sign of gradient.
A1 oe: \(y = \dfrac{1}{2}x\) or \(2y = x\)
| Scheme | Marks |
|---|---|
| \(xy = 16\) or \(y = \dfrac{16}{x}\) or \(x = \dfrac{16}{y}\) | B1 oe |
| (1) |
Notes
B1 oe: Correct Cartesian equation. Accept \(\dfrac{4}{y} = \dfrac{x}{4}\) or \(xy = 4^2\)
| Scheme | Marks |
|---|---|
| Way 1: \(\dfrac{1}{2}x = \dfrac{16}{x}\) \(\{x^2 = 32\}\) Way 2: \(\dfrac{4}{t} = \dfrac{1}{2}(4t)\) \(\{t^2 = 2\}\) Way 3: \(2y = \dfrac{16}{y}\) \(\{y^2 = 8\}\) | M1 |
| \(\left(4\sqrt{2},\ 2\sqrt{2}\right),\ \left(-4\sqrt{2},\ -2\sqrt{2}\right)\) | A1 A1 |
| (3) | |
| (7 marks) |
Notes
M1: Attempts to substitute their \(l\) into either their Cartesian equation or parametric equations of \(H\)
A1: At least one set of coordinates (simplified or un-simplified) or \(x = \pm 4\sqrt{2},\ y = \pm 2\sqrt{2}\)
A1: Both sets of simplified coordinates. Accept written in pairs as \(x = 4\sqrt{2},\ y = 2\sqrt{2}\); \(x = -4\sqrt{2},\ y = -2\sqrt{2}\)