Question

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

Answer 1

Given equation of hyperbola-

$x^2-4y^2=4$

$\Rightarrow \frac{x^2}{4}-\frac{y^2}{1}=1$

here $a=2,b=1$

Centre of the hyperbola $=(0,0)$

Eccentricity $\left(e\right)=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{1}{4}}=\frac{\sqrt{5}}{2}$

Focii of hyperbola $=(\pm ae,0)=(\pm \sqrt{5},0)$

Length of latus rectum $\left(l\right)=\frac{2b^2}{a}=\frac{2\times 1}{2}=1$

Equation of directrices-

$x=\pm \frac{a}{e}\Rightarrow x=\pm \frac{4}{\sqrt{5}}$