Pearson Edexcel IAL Mathematics FP3.2.2 Focus-directrix properties

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

[Maximum number: 4]

The ellipse E has equation

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

(a)

Determine the eccentricity of E

[ 2 ]

$b^{2}=a^{2}\left(1-e^{2}\right) \Rightarrow 4=9\left(1-e^{2}\right) \Rightarrow e=\ldots$

or e.g.
$e=\sqrt{1-\frac{b^{2}}{a^{2}}} \Rightarrow e=\ldots$

Uses a correct formula with a and b
correctly placed to find a value for e

M1

$e=\frac{\sqrt{5}}{3}$

Correct value (or equivalent)
$e= \pm \frac{\sqrt{5}}{3} \text { scores A0 }$

A1

(2)

(b)

Hence, for this ellipse, determine

[ 2 ]

(i)

the coordinates of the foci,

[ 1 ]

$( \pm a e, 0)=( \pm \sqrt{5}, 0) \text { or }\left( \pm 3 \frac{\sqrt{5}}{3}, 0\right)$

Correct foci. Must be coordinates but allow unsimplified and isw if necessary.

Follow through their e so allow for ( ± 3× their e, 0 )

B1ft

(ii)

the equations of the directrices.

The point P lies on E and has coordinates $(3 \cos \theta, 2 \sin \theta)$ .

The line $l_{1}$ is the tangent to E at the point P

[ 1 ]

$x= \pm \frac{a}{e}= \pm \frac{9}{\sqrt{5}} \text { or } x= \pm \frac{3}{\frac{\sqrt{5}}{3}}$

Correct directrices. Must be equations but allow unsimplified and isw if necessary.

Follow through their e so allow for $x= \pm 3$ /their e

B1ft

(2)

Special case:

Use of $a^{2}$ for a and $b^{2}$ for b consistently scores M0A0 in (a) and B1ft B1ft in (b)

This gives $e=\frac{\sqrt{65}}{9},( \pm \sqrt{65}, 0), x= \pm \frac{81}{\sqrt{65}}$

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FP3.2.2 - Focus-directrix properties question 1

Question (a)

Question (b)

Question (i)

Question (ii)