A geometrical problem that arises in physics and engineering is to find the curves that in...

A geometrical problem that arises in physics and engineering is to find the curves that intersect a given family of curves orthogonally. (That is, the tangent lines are perpendicular at the intersection points.) These curves are called the orthogonal trajectories of the given family of curves. For instance, consider the family of circles x2 + y2 = a2 where a is a constant. From the geometry of circles, we can see that the orthogonal trajectories of this family of circles consist of the lines through the origin. Another way of obtaining this that works more generally for other families of curves involves using differential equations.

a) Show that the orthogonal trajectories of the family of circles x2 + y2 – a2 satisfy the differential equation

(Hint: Recall that the slopes of perpendicular lines are the negative reciprocals of each other.)

---

b) Solve the differential equation in part (a) to determine the orthogonal trajectories of the family of circles x2 + y2 = a2.