FP3 June 2010 Q1
1. The line \(x = 8\) is a directrix of the ellipse with equation \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \quad a > 0,\ b > 0,\]
and the point \((2, 0)\) is the corresponding focus.
Find the value of \(a\) and the value of \(b\). (5)
| Scheme | Marks |
|---|---|
| \(\pm\dfrac{a}{e} = 8,\quad \pm ae = 2\) | B1, B1 |
| \(\dfrac{a}{e} \times ae = a^2 = 16\) \(a = 4\) | B1 |
| \(b^2 = a^2(1 - e^2) = a^2 - a^2e^2\) \(\Rightarrow b^2 = 16 - 4 = 12\) | M1 |
| \(\Rightarrow b = \sqrt{12} = 2\sqrt{3}\) | A1 |
| (5) | |
| (5 marks) |