Concept explainers
(a)
To show: The equality of the tangents
(a)
Explanation of Solution
At the points
The arbitrary point P on the curve is on the surface of the cylinder and lies outside the spheres. The point
Since, the points
Recall the fact that, the set of all tangents to a sphere from a point outside of it are equal in length.
Hence by the fact that, the tangents to the sphere from a point outside of it are equal. Therefore,
Similarly, the points
Hence by the fact that, the tangents to the sphere from a point outside of it are equal. Therefore,
Therefore, it is shown that,
(b)
To explain: The reason why
(b)
Explanation of Solution
Since,
Recall the definition of equator of a sphere that the equator of a sphere is a great circle which divides the sphere in to two equal halves.
By the definition of equator of a sphere the vertical distance between
(c)
To show: The addition of tangent lines
(c)
Explanation of Solution
Definition used:
Equator of a sphere:
“The equator of a sphere is a great circle which divides the sphere in to two equal halves”.
From part (b),
From part (a),
Substitute
Thus, it is shown that
(d)
To conclude: The curve is an ellipse with foci
(d)
Explanation of Solution
Definition used:
“If
From section (c), it is true that
That is,
By the definition stated above, the curve is an ellipse.
Thus, it is concluded that the curve is an ellipse with foci
Chapter 11 Solutions
Precalculus: Mathematics for Calculus - 6th 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