If a hyperbola has length of its conjugate axis equal to 5 and the distance between its fo…

Question

If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is:

Accepted Answer

$2b=5\Rightarrow b=\dfrac52$; $2ae=13$. From $b^2=a^2(e^2-1)=(ae)^2-a^2$:
$$\frac{25}4=\left(\frac{13}2\right)^2-a^2\Rightarrow a^2=36\Rightarrow a=6.$$

$$e=\frac{13/2}6=\frac{13}{12}.$$

Solutions · 1