Find Canonical equation of a Hyperbola

Question

Find Canonical equation of a Hyperbola given one focus $(1,3)$ and asymptotes. First asymptote is $x=y$. Second asymptote is $x+y=2$.

Confused with focus.

Thanks!

Answer 1

The answer is $$ (y-1)^2 -(x-1)^2=4$$

The intersection of asymptotes is $(1,1)$ which is the center. The hyperbola is passing through $(1,3)$ therefore we can write the equation $$ (y-1)^2 -(x-1)^2=b$$and solve for $b.$