Cartesian equation of the hyperbola?

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1

How do you find t at a given point of an ellipse?

Equate \cos t=\frac xa and \sin t=\frac yb and choose the correct quadrant.

Why does a line parallel to an asymptote meet a hyperbola once?

The x^2 terms cancel when substituted, leaving a linear equation.

The ellipse and hyperbolaEdexcel International A Level Further Maths: Flashcards

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Cartesian equation of the ellipse centred at the origin?

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Cartesian equation of the ellipse centred at the origin?: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Parametric equations of the ellipse?: $x=a\cos t$ , $y=b\sin t$
Which identity links the parametric and Cartesian forms of the ellipse?: $\cos^2t+\sin^2t=1$
Cartesian equation of the hyperbola?: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Parametric equations of the hyperbola using $\sec$ and $\tan$ ?: $x=a\sec t$ , $y=b\tan t$
Which identity is used for the $\sec$ / $\tan$ form?: $1+\tan^2t=\sec^2t$
Parametric equations of the hyperbola using $\cosh$ and $\sinh$ ?: $x=a\cosh t$ , $y=b\sinh t$
Which identity is used for the $\cosh$ / $\sinh$ form?: $\cosh^2t-\sinh^2t=1$
Which branch does $x=a\cosh t$ , $y=b\sinh t$ trace?: The right branch ( $x\ge a$ ).
How do you obtain the left branch with $\cosh$ and $\sinh$ ?: $x=-a\cosh t$ , $y=b\sinh t$ .
Asymptotes of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ?: $y=\pm\frac bax$
Where does the ellipse cross the axes?: $(\pm a,0)$ and $(0,\pm b)$ .
How do you find $t$ at a given point of an ellipse?: Equate $\cos t=\frac xa$ and $\sin t=\frac yb$ and choose the correct quadrant.
Why does a line parallel to an asymptote meet a hyperbola once?: The $x^2$ terms cancel when substituted, leaving a linear equation.

Exam questions on The ellipse and hyperbola

The ellipse $E$ has parametric equations $x=5\cos t$ , $y=3\sin t$ , for $0\le t<2\pi$ .
Find the value of $t$ , for $0\le t<2\pi$ , at the point $\left(4,-\frac95\right)$ on $E$ . Give your answer to 3 significant figures.2 marks
The hyperbola $H$ has Cartesian equation $\frac{x^2}{9}-\frac{y^2}{16}=1$ .
Show that $x=3\cosh t$ , $y=4\sinh t$ satisfies the Cartesian equation of $H$ .2 marks
The ellipse $E$ has Cartesian equation $\frac{x^2}{16}+\frac{y^2}{4}=1$ .
Write down parametric equations for $E$ . Hence find the exact value of the parameter $t$ , for $0<t<\frac{\pi}{2}$ , at the point $\left(2\sqrt2,\sqrt2\right)$ on $E$ .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).