$x^2 – y^2 = -1$ . 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).
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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.
martini
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Why dividing each factor by 2, means a rotation of coordinate system by 45 degrees? – Al.Ka Apr 07 '15 at 12:16
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The $z = 0$-line, that is the "$w$-axis", is the line, where $x=y$, which is a $45^°$ rotated $x$-axis. The $\sqrt 2$-factors are here to make the transformation lenght preserving (we want the point with $(z,w)$-corrdinates $(1,0)$ to have distance 1 from the origin. – martini Apr 07 '15 at 12:18