Question

If the lengths of the transverse axis and the latus rectum of a hyperbola are $6$ and $\frac{8}{3}$ respectively, then the equation of the hyperbola is ________.

• A. $4x^2-9y^2=72$
• B. $4x^2-9y^2=36$
• C. $9x^2-4y^2=72$
• D. $9x^2-4y^2=36$

Answer 1

Answer: (B)

$4x^2-9y^2=36$

Let the equation of the given hyperbola be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

Then the lengths of its latus rectum and its transverse axis are $\frac{2b^2}{a}$ and $2a$ respectively.

So, we have $2a=6$. That is, $a=3$.

Also, $\frac{2b^2}{a}=\frac{8}{3}$

i.e, $b^2=\frac{4a}{3}$

i.e, $b^2=4$

So, the equation of the hyperbola is $\frac{x^2}{9}-\frac{y^2}{4}=1.$

That is, $4x^2+9y^2=36.$