Edexcel A-Level Mathematics A2 Fp3 2 2 Focus Directrix Properties Questions

Practise using focus-directrix properties of ellipses and hyperbolas to find eccentricity, foci, directrices and curve equations.

Exam points

calculate foci and directrices from a, b and eccentricity data
link PS² and PM² through the focus-directrix definition to prove a curve equation
use b² = a²(1-e²) or b² = a²(e²-1) to find e or the hyperbola equation

[Maximum number: 4]

The ellipse E has equation

\frac{x^{2}}{9}+\frac{y^{2}}{4}=1

(a)

Determine the eccentricity of E

[ 2 ]

$b^2=a^2\left(1-e^2\right)\Rightarrow4=9\left(1-e^2\right)$

$e=\frac{\sqrt5}{3}$

(b)

Hence, for this ellipse, determine

[ 2 ]

(i)

the coordinates of the foci,

[ 1 ]

$(\pm ae,0)=(\pm\sqrt5,0)$

(ii)

the equations of the directrices.

[ 1 ]

$x=\pm\frac{a}{e}=\pm\frac{9}{\sqrt5}$

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