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 ofcircles 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 hyperbolasx 2 = y2 = k (see Figure 2.11).

SolutionStep 1In this problem, we have to show that the curves orthogonal to the family F (x,y) = k satisfy the differential equation Step 2Here, we have to find the perpendicular slope of orthogonal curve If the curve is perpendicular then the slope is Rewrite the above equation where Hence, it is proved that the differential equation will be Step 2Let us assume that Now, differentiate with respect to y and Now, the differential equation will be Divide by 2 on both...