Question

The eccentricity of the hyperbola with latus rectum $12$ and semi-conjugate axis $2\sqrt{3}$ , is

• A. $3$
• B. $\sqrt{\frac{3}{2}}$
• C. $2\sqrt{3}$
• D. 2

Answer 1

Answer: (D)

2

Answer 2

Let equation of hyperbola be.
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
Length of latusrectum, $\frac{2b^2}{a} = 12$
and length of semi-coqjugate axis, $b = 2\sqrt{3}$
$\therefore \frac{2(2\sqrt{3})^2}{a} = 12$
$\Rightarrow a = 2$
$\therefore$ Eccentricity$e = \sqrt{1+ \frac{b^2}{a^2}}$
$= \sqrt{1+ \frac{\left(2\sqrt{3}\right)^{2}}{\left(2\right)^{2}}}$
$= \sqrt{1 + \frac{12}{4}} =2$