Concept explainers
To find : the orthogonal trajectories of the family of curves and draw members of family by using graphing device.
Answer to Problem 30E
Thus the orthogonal trajectories are the family of
Explanation of Solution
Given information :
The equation is
Calculation : Since the orthogonal trajectories to this family are those curves whose tangent lines are orthogonal to the tangent lines of the family, we should determine the slopes of the tangent lines to the given family of curves. To do so, differentiate implicitly:
The differential equation depends on k but it is needed an equation that is valid for all values of k simultaneously. To eliminate k , substitute
Since the orthogonal trajectories should have perpendicular tangent lines, their slopes will be negative reciprocals of the above. In other words, the orthogonal trajectories of the given family are the solutions of the differential equation
Therefore, the orthogonal trajectories must satisfy the differential equation,
The differential equation is separable, so to solve it
Where c is an arbitrary positive constant.
Thus the orthogonal trajectories are the family of
For members of family substitute c by different values ,
Using implicit plotting capability of a CAS to graph the curve
The graph is obtained as :
Chapter 7 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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