Orthogonal trajectory

An orthogonal trajectory intersects every regular member of a specified family of plane curves at a right angle. If eliminating the family parameter gives its tangent slope $dy/dx=F(x,y)$, the dot product of the two tangent directions gives the orthogonal slope $dy/dx=-1/F(x,y)$ wherever both slopes are finite and nonzero. Vertical and horizontal tangent directions are interpreted geometrically. For the family $y=c{x}^n$ with $n>0$, eliminating $c$ gives $F=ny/x$, and the orthogonal trajectories satisfy $x^2+n{y}^2=C$. For $n=2$ these are ellipses; for $n=1$ they are circles.