Question

The eccentricity of the hyperbola $4x^2-9y^2=2ax+b^2$ is

• A. $\frac{a}{b}$
• B. $\frac{\sqrt{b}}{a}$
• C. $\frac{\sqrt{13}}{2}$
• D. $\frac{\sqrt{13}}{3}$

Answer 1

Answer: (D)

$\frac{\sqrt{13}}{3}$

$4x^2-9y^2=2ax+b^2$
$4(x^2-\frac{a}{2}x)-9y^2=b^2$
$4(x^2-2\cdot \frac{a}{4}x+\frac{a^2}{16})-9y^2=b^2-\frac{a^2}{4}$
$4{(x-\frac{a}{4})}^2-9y^2=b^2-\frac{a^2}{4}=k$ (say)
$\frac{{(x-\frac{a}{4})}^2}{k/4}-\frac{y^2}{k/9}=1$
Now assuming $k>0$
Eccentricity of the given hyperbola is $=\sqrt{1+\frac{k/9}{k/4}}=\sqrt{1+\frac{4}{9}}=\frac{\sqrt{13}}{3}$