# For f(x, y)=x-y, how do you prove that the equation f(x, y)= x f(y,x) represents a hyperbola? find its asymptotes?

Oct 22, 2016

The graph is two lines.

#### Explanation:

Substituting $f \left(x , y\right) = x - y$ and $f \left(y , x\right) = y - x$, we get

$x - y = x \left(y - x\right)$

$\implies x - y = x y - {x}^{2}$

$\implies x y + y = {x}^{2} + x$

$\implies y \left(x + 1\right) = x \left(x + 1\right)$

We'll consider two cases, now:

Case 1: $x \ne - 1$

Then we can divide both sides by $x - 1$ to get

$y = x$

Thus, for $x \ne - 1$, the graph matches the line $y = x$.

Case 2: $x = - 1$

Then $y \left(- 1 + 1\right) = - 1 \left(- 1 + 1\right)$

$\implies 0 = 0$

As this is a tautology, $\left(- 1 , y\right)$ is part of the graph for all $y \in \mathbb{R}$. This gives us a vertical line $x = - 1$.

Taken together, our graph becomes two lines: the line with slope $1$ passing through the origin, and the vertical line $x = - 1$.