Concept explainers
To show: One family of the curves are orthogonal trajectories to the other family.
Explanation of Solution
Given:
The family equations
Derivative rules:
Chain rule:
Proof:
The equation of the curve is
Differentiate implicitly with respect to x.
Apply the chain rule and simplify the terms,
Thus, the slope of the tangent to the circle is
Suppose that the circle and straight line are intersect at the point
The slope of the tangent to
The given straight line is passing through point
Clearly, the straight line passing through the points
The slope of the tangent line computed as follows,
Thus, the slope of the straight line is
Note: The two tangent lines are orthogonal if the product of their slopes is -1.
Hence, the family of the curves
Graph:
The family of the circles and straight lines as shown below in Figure 1.
From Figure 1, it is observed that the family of the circle and family of the straight line are orthogonal trajectories.
Chapter 3 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning