A Family of Equilateral Hyperbolas whose asymptotes are the coordinate axes. Obtain the differential equation.

Question

I searched the equation for this question and it's xy=c^2, but I don't know why is that? Please explain further. I've been solving this for so many hours.

Answer 1

We differentiate this equation by x: y + xy' = 0. This is differential equation of this family of hyperbolas,

I think, that C² ≥ 0, xy ≥ 0. So we consider family of hyperbols in I and III quadrant.

Let do step -by-step: y = C²/x; y' = - C²/x^2.

^{Now because C^2 = xy, we get y' = - y/x}