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

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