If the eccentricity and length of latus rectum of a hyperbola are 3 13 and 3 10 units respectively, then what is the length of the transverse axis?

Question

If the eccentricity and length of latus rectum of a hyperbola are 3 13 ​ ​ and 3 10 ​ units respectively, then what is the length of the transverse axis?

TestHub · Accepted Answer

Length of latus rectum of a hyperbola is $\frac{2 b ^{2}}{a}$ where $a$ is the half of the distance between two vertex of the hyperbola.
Latus rectum $= \frac{2 b ^{2}}{a} = \frac{10}{3}$ or, $b^{2} = \frac{5 a}{3}$ ...(i)
In case of hyperbola, $b^{2} = a^{2} (e^{2} - 1)$ ...(ii)
Putting value of $b^{2}$ from equation (i) and $e =$
$\frac{13}{3}$ in equation (ii), $\frac{5 a}{3} = a^{2} (\frac{13}{9} - 1)$ or,
$\frac{5 a}{3} = \frac{4 a ^{2}}{9}$
$\Rightarrow 4 a^{2} - 15 a = 0$ or, $a (4 - 15 a) = 0$
$a \neq = 0$ , hence, $a = \frac{15}{4}$

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