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To get the asymptotes equations, REPLACE the "1" in the right side by "0"
x^2 - y^2 = 0
and factor the left side
(x-y)*(x+y) = 0.
The equations
x - y = 0
and
x + y = 0
are the equation's of the asymptotes, in this case.
Solved, answered and explained.