I know this is probably a super easy problem, but I'm not getting it.

For problems 15-20, find the equation of the orthogonal trajectories to the given family of curves.

x^{2}+4y^{2}=c

I used implicit differentiation and I got dy/dx=-x/4y

I know the slope of the orthagonal trajectory curve will be dy/dx=-1/f(x,y) which is dy/dx=4y/x

integrating that equation gave me y=x^{4}+k but the answer in the back of my textbook is y=kx^{4 }not sure why the constant is in the wrong place