Suppose x and y are very, very large. Then that "1" on the right hand side of
[tex]-\frac{x^2}{4}+ \frac{y^2}{8}= 1[/tex]
is very very small compared to the other two terms so the equation is approximately
[tex]-\frac{x^2}{4}+ \frac{y^2}{8}= 0[/tex]
Of course, [tex]-\frac{x^2}{4}+ \frac{y^2}{8}[/tex]
factors as
[tex](\frac{x}{2}+ \frac{y}{\sqrt{8}})(-\frac{x}{2}+\frac{y}{\sqrt{8}})[/tex]
so for large x,y your equation is approximately
[tex](\frac{x}{2}+\frac{y}{2\sqrt{2}})(-\frac{x}{2}+\frac{y}{2\sqrt{2}})= 0[/tex]
Certainly if "ab= 0" then either a= 0 or b= 0. That gives you two linear equations whose graphs are close to the hyperbola for large x,y: the asymptotes.
