Find the equation of the hyperbola?

Question

The hyperbola being an orthogonal parabola, for which $(-1,2)$ is a focal point and $x-y+1=0$ is an asymptote.

If I have the equation for the asymptote $y=x+1$ is the center $(0,1)$?

I do not know where to proceed next.

Answer 1

No. The center is $(-1,0)$.

Since the hyperbola being an orthogonal parabola, we have $$\frac{(x-p)^2}{a^2}-\frac{(y-q)^2}{b^2}=-1.$$ Since $(−1,2)$ is a focal point, we have $$-1=p, 2=\sqrt{a^2+b^2}+q.$$ Since $x−y+1=0\iff y=x+1$ is an asymptote, we have $$1=\frac ba, 1=-\frac{bp}{a}+q$$ Now we have $$q=0, p=-1, a=b, a^2=2.$$ Hence, the center is $(p,q)=(-1,0)$.