Find the equation of the equilateral hyperbola given one of its foci points $(1;1)$ and an asymptote $x+y=0$

Question

If the asymptote is $y=-x$, then doesn't it mean that the other one is $y=x$?

But does't that also mean that the foci points are $(c_1,0)$, $(c_2,0)$?

But then how is it $(1,1)$?

Answer 1

HINT
The picture shows a rectangular (equilateral) hyperbola, its asymptotes and one of the foci.
In this particular case we see a square with vertices: the focus, its orthogonal projection on one asymptote, the center of the hyperbola and projection of the focus on other asymptote.

EDIT

This new picture shows the given point $(1,1),$ the asymptote $y+x=0$ and one of convenient hyperbolas.
The second hyperbola corresponds to the green square. Its center is $(1,-1)$ and the second focus is the small red point bellow.

Answer 2

Hope the following shifted rectangular hyperbola addresses all questions: