Answer
$\frac{y^2}{3}-\frac{x^2}{3}=1$
Work Step by Step
Asymptotes:
$y=±\frac{a}{b}x$ (vertical transverse axis)
$y=±\frac{b}{a}x$ (horizontal transverse axis)
$y=±x$
$a=b$
The point $(1,2)$ is above both $y=x$ and $y=-x$ because $2\gt1$ (see the graph). We have a hyperbola with vertical transverse axis:
Use the point $(1,2)$.
Hyperbola with vertical transverse axis:
$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
$\frac{2^2}{a^2}-\frac{1^2}{a^2}=1$
$\frac{3}{a^2}=1$
$a^2=3$
$b^2=3$
Finally:
$\frac{y^2}{3}-\frac{x^2}{3}=1$
