What kind of line does this equation represent?

Question

$x^2 – y^2 = -1$ . I know it is a hyperbola, but i want to know to reach this conclusion, (sorry for the symbols but I do not know how to use MathJax).

Answer 1

Let's rewrite your equation as $$ (x+y)(x-y) = -1 $$ If we make a change of coordinates by setting $$ z := \frac{x-y}{\sqrt 2}, \quad w := \frac{x+y}{\sqrt 2} $$ (think of it as a rotation of the coordinate system by $45^°$), we have in our new coordinates $$ wz = -2 \iff z = -\frac 2w, $$ which describes a hyperbola.