An orthogonal trajectory is a family of curves that intersect another family of curves at right angles (orthogonally) at every point of intersection. In other words, when two curves from different families intersect, their tangent lines at the point of intersection are perpendicular.
Orthogonal trajectories are useful in various applications, including physics, geometry, and engineering. For example, they can describe the behavior of electric and magnetic fields, where field lines and equipotential lines are orthogonal trajectories.
To find the orthogonal trajectories of a given family of curves, one can first find the differential equation representing the given family, then swap the roles of dy/dx (the derivative of y with respect to x) with -dx/dy (the negative reciprocal of the derivative of x with respect to y) to obtain the differential equation for the orthogonal trajectories.