Concept explainers
Show that the ellipse x2/a2 + y2/b2 = 1 and the hyperbola x2/A2 – y2/ B2 = 1 are orthogonal trajectories if A2 < a2 and a2 – b2 = A2 + B2 (so the ellipse and hyperbola have the same foci).
To show: The ellipse and the hyperbola are orthogonal trajectories if
Explanation of Solution
Given:
The equation of the ellipse
The equation of the hyperbola
Derivative rules:
Chain rule:
Proof:
Consider equation of the ellipse
Differentiate implicitly with respect to x,
Apply the chain rule and simplify the terms,
Thus, the derivative of the equation of the ellipse is
That is, the slope of the tangent to equation of the ellipse is
Consider equation of the ellipse
Differentiate implicitly with respect to x,
Apply the chain rule and simplify the terms,
Thus, the derivative of the equation of the ellipse is
That is, the slope of the tangent to equation of the hyperbola is
Suppose that
It is required to prove that the equations are orthogonal trajectories.
That is, the product of their slope is -1.
Subtract the equation (2) from the equation (1),
Since
Substitute
Substitute the value
Therefore, the ellipse and the hyperbola are orthogonal trajectories if
Hence the required result is proved.
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