Question

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point $(3\sqrt{5},1)$ and the length of its latus rectum is $\frac{4}{3}$ units. The length of the conjugate axis is.

• A. $2units$
• B. $3units$
• C. $4units$
• D. $5units$

Answer 1

The correct option is C $4units$

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through $(3\sqrt{5},1)$

$⟹\frac{45}{a^2}-\frac{1}{b^2}=1$ ..... $\left(i\right)$

The latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{2b^2}{a}=\frac{4}{3}$

$⟹b^2=\frac{2a}{3}$

$⟹\frac{45}{a^2}-\frac{3}{2a}=1$ .... From $\left(i\right)$

$⟹a=\frac{-30}{4},6$

But $a>0,$ then $a=6$

$⟹b=2$

Length of conjugate axis is $2b=4$