Question

In the hyperbola $4x^2-9y^2=36$, find lengths of the axes, the co-ordinates of the foci, the eccentricity, and the latus rectum.

• A. $6,4;(\pm \sqrt{13},0);\frac{\sqrt{13}}{3};\frac{8}{3}$
• B. $9,4;(\pm \sqrt{8},0);\frac{\sqrt{8}}{3};\frac{8}{3}$
• C. $9,4;(\pm \sqrt{13},0);\frac{\sqrt{13}}{3};\frac{8}{3}$
• D. $6,4;(\pm \sqrt{8},0);\frac{\sqrt{8}}{3};\frac{8}{3}$

Answer 1

The correct option is A $6,4;(\pm \sqrt{13},0);\frac{\sqrt{13}}{3};\frac{8}{3}$

Given hyperbolas may be written as, $\frac{x^2}{9}-\frac{y^2}{4}=1$
$\Rightarrow a^2=9,b^2=4$
$\therefore$ Eccentricity is, $e=\sqrt{1+\frac{b^2}{a^2}}=\frac{\sqrt{13}}{3}$
Thus, length of axes are $2a$ and $2b\Rightarrow 6$ and $4$
Focus $\equiv (\pm ae,0)=(\pm \sqrt{13},0)$
And length of latus rectum $=\frac{2b^2}{a}=\frac{8}{3}$