A2 October 2021 Q1
1. The ellipse \(E\) has equation
\[\frac{x^2}{36} + \frac{y^2}{20} = 1\]Find
(a) the coordinates of the foci of \(E\), (3)
(b) the equations of the directrices of \(E\). (2)
| Scheme | Marks | AO |
|---|---|---|
| Uses \(b^2 = a^2\left(1 - e^2\right)\) to find a value of \(e\) look for \(20 = 36\left(1 - e^2\right)\) | M1 | 1.1b |
| \(e = \dfrac{2}{3} \Rightarrow\) foci are \(\left(\pm 6 \times \text{“their } e\text{”}, 0\right)\) | dM1 | 1.1b |
| Foci are \((\pm 4, 0)\) | A1 | 1.1b |
| (3) |
Notes
M1: Uses \(b^2 = a^2\left(1 - e^2\right)\) to obtain a value of \(e\) (allow if \(-\dfrac{2}{3}\) also given)
dM1: Uses \(a = 6\) and their value of \(e\) with \(0 \lt e \lt 1\), to find at least one focus using \(\left((\pm)ae, 0\right)\)
A1: Correct foci – both required, including \(y\) coordinates.
Alternative
| Scheme | Marks | AO |
|---|---|---|
| Sets up an equation such as \(2\sqrt{p^2 + b^2} = 2a\) where \(p\) is the \(x\) coordinate of the foci \(2\sqrt{p^2 + 20} = 12\) | M1 | 1.1b |
| Solves to find the value of \(p\) | dM1 | 1.1b |
| Foci are \((\pm 4, 0)\) | A1 | 1.1b |
| (3) |
M1: Sets up an equation using total distance from foci to point on ellipse \(= 2a\)
dM1: Solves to find a value for the \(x\) coordinate of the foci
A1: Correct foci – both required, including \(y\) coordinates.
| Scheme | Marks | AO |
|---|---|---|
| Directrices are \(x = (\pm)\dfrac{6}{\text{their } e}\) | M1 | 1.1b |
| \(x = \pm 9\) only | A1 | 1.1b |
| (2) | ||
| (5 marks) |
Notes
M1: Uses \(x = (\pm)\dfrac{a}{e}\) with \(a = 6\) and their \(e\) to attempt directrices.
A1: Correct directrices, both required and no other lines