points (0,1), and (0,-1), where the curve is horizontal.
Given the orientation of the hyperbola, its equation is of the form:
-a x^2 + b y^2 = 1
for some a, b>0. Then the condition that it passes through (0,1) gives:
and as the asymptotes are y=+/- x, we also have a=b, so the hyperbola
x^2 - y^2 = 1.