Question

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?

Answer 1

The graph is two lines.

Explanation:

Substituting `f(x,y) = x-y` and `f(y,x) = y-x`, we get

`x-y = x(y - x)`

`=> x - y = xy - x^2`

`=> xy + y = x^2 + x`

`=> y(x+1) = x(x+1)`

We'll consider two cases, now:

Case 1: `x != -1`

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

`y = x`

Thus, for `x!=-1`, the graph matches the line `y=x`.

Case 2: `x = -1`

Then `y(-1+1) = -1(-1+1)`

`=> 0=0`

As this is a tautology, `(-1, y)` is part of the graph for all `y in RR`. 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`.