Find the orthogonal trajectories of the family of curves.
I don't know how to begin this!
differentiate your curve: $\displaystyle x+2yy'=0.$ replace $\displaystyle y'$ with $\displaystyle \frac{-1}{y'}$ to get the differential equation of the orthogonal trajectories: $\displaystyle \frac{y'}{y}=\frac{2}{x}.$ solve the equation to get: $\displaystyle y=cx^2.$