Question

The foci of the hyperbola $4 x^{2}-9 y^{2}-1=0$ are

• A. $\left(\pm\sqrt{13,}0\right)$
• B. $\left(\pm\sqrt{\frac{13}{6},}0\right)$
• C. $\left(0,\pm\sqrt{\frac{13}{6}}\right)$
• D. None of these

Answer 1

Answer: (B)

The given equation of hyperbola can be re-written as,
$\frac{ x ^{2}}{\left(\frac{1}{2}\right)^{2}}-\frac{ y ^{2}}{\left(\frac{1}{3}\right)^{2}}=1$
So, the hyperbola has $x$-axis as the transverse axis, with $a=\frac{1}{2}$
and $b=\frac{1}{3}$
And, $e=\sqrt{1+\frac{b^{2}}{a^{2}}}$
$=\sqrt{1+\frac{4}{9}}=\frac{\sqrt{13}}{3}$
$\therefore $ Focii $=(\pm ae , 0)=\left(\pm \frac{\sqrt{13}}{6}, 0\right)$