Question

If the latus rectum of an hyperbola be $8$ and eccentricity is $\frac{3}{\sqrt{5}}$ then the equation of the hyperbola can be

• A. $4x^2-5y^2=100$
• B. $5x^2-4y^2=100$
• C. $4x^2+5y^2=100$
• D. $5x^2+4y^2=100$

Answer 1

Answer: (A)

$4x^2-5y^2=100$

Give eccentricity of the hyperbola is, $e=\frac{3}{\sqrt{5}}\Rightarrow \frac{b^2}{a^2}=\frac{4}{5}..\left(1\right)$
And latus rectum is $=2\frac{b^2}{a}=8\Rightarrow \frac{b^2}{a}=4\dots \left(2\right)$
By $\left(2\right)/\left(1\right)$ $a=5\therefore b^2=20$
Hence required hyperbola is, $\frac{x^2}{25}-\frac{y^2}{20}=1$
$\Rightarrow 4x^2-5y^2=100$
Hence, option 'A' is correct.