Question

find the orthogonal trajectories of the family of hyperbolas xy=c (c not= 0 )

Answer 1

$$Solution$$

From the given family of curves, we find a differential equation the curves all satisfy,

$$xy=C \implies y'=-\frac{C}{x^2}=-\frac{y}{x}$$

Letting $f(x,y)=-\frac{y}{x}$, we know the orthogonal trajectories are the curves which satisfy a differential equation

$$y'=-\frac{1}{f(x,y)}\implies y'=\frac{x}{y}$$

Therefore, the orthogonal trajectories are the curves,

$$y'=\frac{x}{y} \implies yy'=x \implies \frac12y^2 =\frac12x^2+C\\ \implies y^2-x^2=C$$

where $C$ is an arbitrary constant.