Question

Find the eccentricity, coordinates of foci, length of latus recturm and equation of directrix of the hyperbola $3x^2-y^2=4$.

Answer 1

Equation of the given hyperbola in standard form is:

$\frac{x^2}{\frac{4}{3}}-\frac{y^2}{4}=1$

Comparing with the form:

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

we get:

$a^2=\frac{4}{3},b^2=4$

Thus, eccentricity:

$e=\sqrt{1+\frac{b^2}{a^2}}=2$

Foci are at: $(\pm ae,0)=(\pm \frac{4}{\sqrt{3}},0)$

Length of latus rectum:

$\frac{2b^2}{a}=4\sqrt{3}$

Equation of directrix:

$x=\pm \frac{a}{e}\Rightarrow x=\pm \frac{1}{\sqrt{3}}$