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?

1 Answer
Oct 22, 2016

Answer:

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#.

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