Question

The eccentricity of the hyperbola whose latus-rectum is half of its transverse axis,

• A. $\frac{1}{\sqrt{2}}$
• B. $\sqrt{\frac{2}{3}}$
• C. $\sqrt{\frac{3}{2}}$
• D. none of these

Answer 1

The correct option is C

$\sqrt{\frac{3}{2}}$

$\text{The length of the latus rectum and the transverse axis are}\frac{2b^2}{a}and2arespectively.$

$\text{According to the given statement,length of the latus rectum is half of its trannsverse axis :}$

$\therefore \frac{2b^2}{a}=\frac{1}{2}\times 2a$

$\Rightarrow \frac{2b^2}{a}=a$

$\Rightarrow 2b^2=a^2$

$Eccebtricity,e=\frac{\sqrt{a^2+b^2}}{a}$

$Substitutingthevalue{b}^2=\frac{a^2}{2},weget$

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

$=\frac{a\sqrt{2}/2}{a}$

$=\sqrt{\frac{3}{2}}$

$\therefore Eccintricityis\sqrt{\frac{3}{2}}$