Question

Find the equation of hyperbolas satisfying the following conditions: $Foci$ at $(0,\pm 6)$ and length of conjugate axis $=2\sqrt{11}$

Answer 1

Since foci is on y-axis

so required equation of hyperbola is

$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

Now, coordinates of foci = $(0,\pm c)$ and given foci is at $(0,\pm 6)$

$\therefore c=6$

Length of conjugate axis = 2b

$2\sqrt{11}=2b$

$\therefore b=\sqrt{11}$

We know that,

$c^2=a^2+b^2$

$6^2=a^2+11$ $\therefore a^2=25$

Thus, required equation of hyperbola, is

$\frac{y^2}{25}-\frac{x^2}{11}=1$