FP3 June 2012 Q1
1. The hyperbola \(H\) has equation \[\frac{x^2}{16} - \frac{y^2}{9} = 1\]
Find
(a) the coordinates of the foci of \(H\), (3)
(b) the equations of the directrices of \(H\). (2)
| Scheme | Marks |
|---|---|
| Uses formula to obtain \(e = \dfrac{5}{4}\) | M1A1 |
| Uses \(ae\) formula | M1 |
| (3) |
Notes
a1M1: Uses \(b^2 = a^2(e^2 - 1)\) to get \(e > 1\)
a1A1: cao
a2M1: Uses \(ae\)
| Scheme | Marks |
|---|---|
| Uses other formula \(\dfrac{a}{e}\) | M1 |
| Obtains both Foci are \((\pm 5, 0)\) and Directrices are \(x = \pm\tfrac{16}{5}\) (needs both method marks) | A1 cso |
| (2) | |
| (5 marks) |
Notes
b1M1: Uses \(\dfrac{a}{e}\)
b1A1: cso for both foci and both directrices. Must have both of the 2 previous M marks may be implicit.