Orthogonal Trajectories. A geometric problem occurring often in engineering is that...

Orthogonal Trajectories. A geometric problem occurring often in engineering is that of finding a family of curves (orthogonal trajectories) that intersects a given family of curves orthogonally at each point. For example, we may be given the lines of force of an electric field and want to find the equation for the equipotential curves. Consider the family of curves described by F (x,y) = k , where k is a parameter. Recall from the discussion of equation (2) that for each curve in the family, the slope is given by

Recall that the slope of a curve that is orthogonal (perpendicular) to a given curve is just the negative reciprocal of the slope of the given curve. Using this fact, show that the curves orthogonal to the family F (x,y) = k satisfy the differential equation

(b) Using the preceding differential equation, show that the orthogonal trajectories to the family of

circles x² +y² = k are just straight lines through the origin (see Figure 2.10).

(c) Show that the orthogonal trajectories to the family of hyperbolas are the hyperbolas

x ² = y² = k (see Figure 2.11).