Question

If the eccentricity and length of latus rectum of a hyperbola are $\frac{\sqrt{13}}{3}$ and $\frac{10}{3}$ units respectively, then find twice the length of the transverse axis.

Answer 1

Answer: 15

$e=\frac{\sqrt{13}}{3}$
$\frac{2b^2}{a}=\frac{10}{3}\Rightarrow \frac{b^2}{a}=\frac{5}{3}\Rightarrow b^2=\frac{5a}{3}$
also $e^2=1+\frac{b^2}{a^2}$
$\Rightarrow \frac{13}{9}=1+\frac{5a/3}{a^2}\Rightarrow \frac{13}{9}=1+\frac{5}{3a}$
$\Rightarrow \frac{4}{9}=\frac{5}{3a}\Rightarrow a=\frac{15}{4}$ .
$\Rightarrow 2\left(2a\right)=4\left(\frac{15}{4}\right)=15$