Where are the ellipse's foci and directrices compared with a?

Foci at ae<a (inside); directrices at \frac aea (outside).

Where are the hyperbola's foci and directrices compared with a?

Foci at aea; directrices at \frac ae<a (between the vertices).

How do you turn PS=e\times PM into an equation?

Square both sides, write PS^2 and PM^2 in x and y, and rearrange.

Which focus goes with which directrix?

(ae,0) with x=\frac ae, and (-ae,0) with x=-\frac ae.

Focus-directrix properties and eccentricityEdexcel International A Level Further Maths: Flashcards

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State the focus-directrix property.

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State the focus-directrix property.: $PS=e\times PM$ , with $M$ the foot of the perpendicular from $P$ to the directrix.
What value of $e$ gives an ellipse?: $0<e<1$
What value of $e$ gives a hyperbola?: $e>1$
Ellipse: relation between $a$ , $b$ and $e$ ?: $b^2=a^2\left(1-e^2\right)$
Hyperbola: relation between $a$ , $b$ and $e$ ?: $b^2=a^2\left(e^2-1\right)$
Foci of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ?: $(\pm ae,0)$ in both cases.
Directrices of the ellipse and hyperbola?: $x=\pm\frac ae$
Where are the ellipse's foci and directrices compared with $a$ ?: Foci at $ae<a$ (inside); directrices at $\frac ae>a$ (outside).
Where are the hyperbola's foci and directrices compared with $a$ ?: Foci at $ae>a$ ; directrices at $\frac ae<a$ (between the vertices).
Eccentricity of $\frac{x^2}{25}+\frac{y^2}{16}=1$ ?: $\frac35$
Eccentricity of $\frac{x^2}{9}-\frac{y^2}{16}=1$ ?: $\frac53$
How do you find $a$ given foci $(\pm c,0)$ and $e$ ?: $ae=c$ , so $a=\frac ce$ .
How do you turn $PS=e\times PM$ into an equation?: Square both sides, write $PS^2$ and $PM^2$ in $x$ and $y$ , and rearrange.
Which focus goes with which directrix?: $(ae,0)$ with $x=\frac ae$ , and $(-ae,0)$ with $x=-\frac ae$ .

Exam questions on Focus-directrix properties and eccentricity

The ellipse $E$ has equation $\frac{x^2}{25}+\frac{y^2}{16}=1$ .
Find the equations of the directrices of $E$ .2 marks
The hyperbola $H$ has equation $\frac{x^2}{9}-\frac{y^2}{16}=1$ .
The point $P\left(5,\frac{16}{3}\right)$ lies on $H$ . Verify that $PS=e\times PM$ , where $S$ is the focus $(5,0)$ and $M$ is the foot of the perpendicular from $P$ to the corresponding directrix.2 marks
An ellipse $E$ has its centre at the origin and its foci at $(\pm4,0)$ on the $x$ -axis. Its eccentricity is $\frac23$ .
Find the equation of $E$ in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).