Question

The distance between the foci of a hyperbola is double the distance between its vertices and the length of its conjugate axis is $6$. Then which of the following is/are correct?

• A. length of transverse axis $=\sqrt{3}$ unit
• B. length of latus rectum $=3\sqrt{3}$ unit
• C. length of transverse axis $=2\sqrt{3}$ unit
• D. length of latus rectum $=6\sqrt{3}$ unit

Answer 1

Answer: (C), (D)

The correct options are
C length of transverse axis $=2\sqrt{3}$ unit
D length of latus rectum $=6\sqrt{3}$ unit

Let the equation of the hyperbola is
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Given
$2ae=2\cdot 2a\Rightarrow e=2$
and $2b=6\Rightarrow b=3$
we know that
$\Rightarrow e=2=\sqrt{1+\frac{b^2}{a^2}}=\sqrt{1+\frac{9}{a^2}}$
$\Rightarrow a=\sqrt{3}$
So, length of transverse axis
$=2a=2\sqrt{3}$ unit
and length of latus rectum
$=\frac{2b^2}{a}=6\sqrt{3}$ unit