Chapter 3.1, Problem 41ES

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of intersection, and two families of curves are said to be orthogonal trajectories of one another if each member of one family is orthogonal to each member of the other family. This terminology is used in these exercises.

The accompanying figure on the next page shows some typical members of the families of circles $x^2+{\left(y-c\right)}^2=c^2$ (black curves) and ${\left(x-k\right)}^2+y^2=k^2$ (gray curves). Show that these families are orthogonal trajectories of one another.