Q.Find the centre, foci, eccentricity, equation of the directrices, length of the latus rectum of the hyperbola. $x^2-4y^2=4$.

Rewrite in standard form $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ and read off each quantity from $a$, $b$, $c=\sqrt{a^2+b^2}$.

Divide $x^2-4y^2=4$ by $4$: $\dfrac{x^2}{4}-\dfrac{y^2}{1}=1$

So $a^2=4$ ($a=2$), $b^2=1$ ($b=1$).

$c^2=a^2+b^2=5 \Rightarrow c=\sqrt5$

• Centre: $(0,0)$
• Foci: $(\pm\sqrt5,0)$
• Eccentricity: $e=\dfrac{c}{a}=\dfrac{\sqrt5}{2}$
• Directrices: $x=\pm\dfrac{a}{e}=\pm\dfrac{a^2}{c}=\pm\dfrac{4}{\sqrt5}$ …