State the Cartesian and parametric equations of the hyperbola.

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1; x=a\sec t, y=b\tan t (or a\cosh t, b\sinh t).

State the foci and directrices of the ellipse and hyperbola.

Foci (\pm ae,0); directrices x=\pm\frac ae.

State the focus-directrix property.

SP=e\times PM: distance to the focus is e times the distance to the directrix.

State the tangent to the ellipse at (a\cos t,b\sin t).

\frac{x\cos t}{a}+\frac{y\sin t}{b}=1.

What conic is the locus SP=e\times PM for e1?

A hyperbola (e<1 gives an ellipse and e=1 a parabola).

Ellipse and hyperbolaEdexcel A-Level Further Maths: Flashcards

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State the Cartesian and parametric equations of the ellipse.

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State the Cartesian and parametric equations of the ellipse.: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ; $x=a\cos t$ , $y=b\sin t$ .
State the Cartesian and parametric equations of the hyperbola.: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ; $x=a\sec t$ , $y=b\tan t$ (or $a\cosh t$ , $b\sinh t$ ).
What identity justifies $x=a\sec t$ , $y=b\tan t$ ?: $\sec^2t-\tan^2t=1$ .
Relate $a$ , $b$ and $e$ for an ellipse.: $b^2=a^2(1-e^2)$ , with $0<e<1$ .
Relate $a$ , $b$ and $e$ for a hyperbola.: $b^2=a^2(e^2-1)$ , with $e>1$ .
State the foci and directrices of the ellipse and hyperbola.: Foci $(\pm ae,0)$ ; directrices $x=\pm\frac ae$ .
State the focus-directrix property.: $SP=e\times PM$ : distance to the focus is $e$ times the distance to the directrix.
Find $e$ for $\frac{x^2}{25}+\frac{y^2}{16}=1$ .: $16=25(1-e^2)$ , so $e=\frac35$ .
State $\frac{dy}{dx}$ for the ellipse.: $-\frac{b^2x}{a^2y}$ .
State $\frac{dy}{dx}$ for the hyperbola.: $\frac{b^2x}{a^2y}$ .
State the tangent to the ellipse at $(a\cos t,b\sin t)$ .: $\frac{x\cos t}{a}+\frac{y\sin t}{b}=1$ .
State the condition for $y=mx+c$ to touch the ellipse.: $c^2=a^2m^2+b^2$ .
State the condition for $y=mx+c$ to touch the hyperbola.: $c^2=a^2m^2-b^2$ .
What conic is the locus $SP=e\times PM$ for $e>1$ ?: A hyperbola ( $e<1$ gives an ellipse and $e=1$ a parabola).

Exam questions on Ellipse and hyperbola

The ellipse $E$ has equation $\frac{x^2}{25}+\frac{y^2}{16}=1$ .
The point $P$ on $E$ has $x$ -coordinate $\frac{10}{3}$ . Use the focus-directrix property to find the distance from $P$ to the focus $(3,0)$ .2 marks
The hyperbola $H$ has equation $\frac{x^2}{9}-\frac{y^2}{16}=1$ , with parametric equations $x=3\sec t$ , $y=4\tan t$ .
Find the exact gradient of $H$ at the point $P\left(6,4\sqrt3\right)$ .2 marks
The hyperbola $H$ has equation $\frac{x^2}{4}-\frac{y^2}{9}=1$ , with parametric equations $x=2\sec t$ , $y=3\tan t$ .
The line $y=2x+k$ is a tangent to $H$ . Find the possible values of $k$ .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).