Question

The equation of the hyperbola with eccentricity $\frac{3}{2}$ and foci at $(\pm 2,0)$

• A. $\frac{x^2}{4}-\frac{y^2}{5}=\frac{4}{9}$
• B. $\frac{x^2}{9}-\frac{y^2}{9}=\frac{4}{9}$
• C. $\frac{x^2}{4}-\frac{y^2}{9}=1$
• D. $Noneofthese$

Answer 1

The correct option is D $Noneofthese$

Eccentricity $\left(e\right)=\frac{3}{2}$

$foci=(\pm \frac{3}{2},0)$

$foci=\frac{3}{2}$$\Rightarrow ae=\frac{3}{2}$

$\Rightarrow a\frac{3}{2}=\frac{3}{2}$ $\Rightarrow a=1$

Putting $e=\sqrt{\frac{a^2+y^2}{a^2}}$

$\Rightarrow {\left(\frac{3}{2}\right)}^2=\frac{a^2+b^2}{a^2}$

$\Rightarrow \frac{9}{4}=\frac{1+b^2}{1}$

$\Rightarrow 4+r{b}^2=9$

$\Rightarrow 4b^2=5$

$b^2=\frac{5}{4}$

Equation of hyperbola

$=\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\Rightarrow \frac{x^2}{1}-\frac{4y^2}{5}=1$

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