Orthogonal trajectories are curves that intersect a given family of curves at right angles. Method: (1) Find the DE of the given family (eliminate the parameter). (2) Replace dy/dx by -dx/dy (negative reciprocal of slope). (3) Solve the new DE. Example: for y = $cx^2$ (family of parabolas), differentiate: y' = 2cx = 2y/x. Replace: -dx/dy = 2y/x, so x dx = -2y dy, giving $x^2$/2 = -$y^2$ + C, i.e., $x^2$ + 2$y^2$ = K (family of ellipses). This is a common JEE application that combines DE formation and solution.