Math Help - hyperbola

1. hyperbola

Consider the hyperbola x^2/a^2+y^2/b^2 = D (where D is the distance between the foci). Without knowing any specific values of x or y calculate the slope of the asymptote.

Thought: b can solved for and expressed as a complex number. Is its length in the complex plane the same as the desired length in the real plane, that is, is the length of the complex vector b the same as the distance from a focal point on the x axis to the asymptote?

First, that is NOT a hyperbola, it is an ellipse. I am going to assume you meant $x^2/a^2- y^2/b^2= D$. For very, very large x and y, $x^2/a^2$ and $y^2/b^2$ will be very, very large compared with D. That means that, "approximately", or, perhaps better, "comparitively", $x^2/a^2- y^2/b^2= (x/a+ y/b)(x/a- y/b)= 0$. That is, the hyperbola will be close to the two lines x/a+ y/b= 0 and x/a- y/b= 0, the asymptotes. What are the slopes of those lines?