Answer
$\frac{y^2}{9}-\frac{x^2}{16}=1$
Work Step by Step
The foci lie on the y-axis. Also, the origin is the center because it is the midpoint between the foci: $\frac{(0,5)+(0,-5)}{2}=(0,0)$. So:
$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
Asymptotes: $y=±\frac{a}{b}x=±\frac{3}{4}x$
$\frac{a}{b}=\frac{3}{4}$
$a=\frac{3}{4}b$
Foci: $F(0,±c)=F(0,±5)$
$c=5$
$c^2=a^2+b^2$
$25=(\frac{3}{4}b)^2+b^2$
$25=\frac{9}{16}b^2+b^2=\frac{25}{16}b^2$
$b^2=16$
$b=4$
$a=\frac{3}{4}b=3$
Finally:
$\frac{y^2}{3^2}-\frac{x^2}{4^2}=1$
$\frac{y^2}{9}-\frac{x^2}{16}=1$
