Question

Equation of the hyperbola with centre $(0,0)$ distance between the foci is $18$ and distance between directrices $8$.

Answer 1

Given hyperbola equation contains: centre $=(0,0)$
Distance between foci$=2ae=18$ where, $a$ is length of semi major axis.
$\Rightarrow ae=9$......(i)
Distance between directrix $=\frac{2a}{e}=8$
$\Rightarrow \frac{a}{e}=4$....(ii)
Multiply equation(i) and (ii), we get
$ae\times \frac{a}{e}=9\times 4$
$\Rightarrow a^2=36$
$\therefore a=6$
From equation (i)$÷$equation(ii), we get $\Rightarrow \frac{ae}{\frac{a}{e}}=\frac{9}{4}$
$\Rightarrow e^2=\frac{9}{4}$
Since, $e^2=1+\frac{b^2}{a^2}$
$\Rightarrow \frac{9}{4}=1+\frac{b^2}{36}$ [Since, a=6]
$\Rightarrow \frac{b^2}{36}=\frac{9}{4}-1$
$\Rightarrow \frac{b^2}{36}=\frac{9-4}{4}$
$\Rightarrow b^2=\frac{5}{4}\times 36$
$\Rightarrow b^2=45$
Thus, required equation of hyperbola would be $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$
Substitute $a^2=36,b^2=45$, we get
$\therefore \frac{x^2}{36}-\frac{y^2}{45}=45$ is the required equation of hyperbola.