Orthogonal trajectories Two curves are orthogonal to each

Chapter , Problem 42

(choose chapter or problem)

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection. A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. Use the following steps to find the orthogonal trajectories of the family of ellipses 2x2 + y2 = a2. a. Apply implicit differentiation to 2x2 + y2 = a2 to show that dy dx = -2x y . b. The family of trajectories orthogonal to 2x2 + y2 = a2 satisfies the differential equation dy dx = y 2x . Why? c. Solve the differential equation in part (b) to verify that y2 = eC_x_ and then explain why it follows that y2 = kx. Therefore, the family of parabolas y2 = kx forms the orthogonal trajectories of the family of ellipses 2x2 + y2 = a2.