asymptote to the curve

Question

the only asymptote to the curve x^3+y^3=3axy is x+y+a=0

Answer 1

Consider a=0, then from both equations, y=-x. So in this case the tangent of the cubic is identical to the line; but it isn't an asymptote.

Now consider a≠0.

(x+y)³ expands to x³+3x²y+3xy²+y³=x³+y³+3xy(x+y).

But x³+y³=3axy, so (x+y)³=3axy+3xy(x+y)=3xy(a+x+y). 

As x (and hence y) becomes large, 3axy becomes relatively small and negligible by comparison so y³≈-x³, implying that y≈-x and x+y→0, therefore (x+y)³→0. Since (x+y)³=3xy(a+x+y)→0, a+x+y→0, making a+x+y=0 an asymptote, since xy→-x², which is never zero, but has a large magnitude as a negative number, when x is large.