FP1 January 2009 Q3
3. The rectangular hyperbola, \(H\), has parametric equations \(x = 5t,\ y = \dfrac{5}{t},\ t \neq 0\).
(a) Write the cartesian equation of \(H\) in the form \(xy = c^2\). (1)
Points \(A\) and \(B\) on the hyperbola have parameters \(t = 1\) and \(t = 5\) respectively.
(b) Find the coordinates of the mid-point of \(AB\). (3)
| Scheme | Marks |
|---|---|
| \(xy = 25 = 5^2\) or \(c = \pm 5\) | B1 |
| (1) |
Notes
(a) \(xy = 25\) only B1, \(c^2 = 25\) only B1, \(c = 5\) only B1
| Scheme | Marks |
|---|---|
| \(A\) has co-ords (5, 5) and \(B\) has co-ords (25, 1) | B1 |
| Mid point is at (15, 3) | M1A1 |
| (3) | |
| [4] |
Notes
(b) Both coordinates required for B1
Add theirs and divide by 2 on both for M1