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Q.
If the eccentricity and length of latus rectum of a hyperbola are $\frac{\sqrt{13}}{3}$ and $\frac{10}{3}$ units respectively, then find the length of the transverse axis.
Conic Sections
Solution:
$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} $
$\Rightarrow \frac{4}{9}=\frac{5}{3 a}$
$\Rightarrow a=\frac{15}{4}$