Concept explainers
(a)
To explain: e a number between 0 and 1.
(a)
Answer to Problem 24WE
Explanation of Solution
Given:
Suppose that
Calculation:
The given equation of the ellipse is,
The eccentricity is e where
Explain why e is a number between 0 and 1.
From the given equation of the ellipse
Hence write as
From this write as
And
Using equation (1),
Dividing all the terms by
By taking square root in all the terms,
The e value cannot be negative since e is between 0 and 1.
Therefore,
Thus proved that e is a number between 0 and 1.
Conclusion:
Therefore,
(b)
To find: the ellipses shown above has the greater eccentricity.
(b)
Answer to Problem 24WE
Therefore the third figure has greater eccentricity
Explanation of Solution
Calculation:
Find that which has greater eccentricity
In the first figure, the length of the major axis is equal to the length of the minor axis.
Therefore,
Now
By squaring in both sides,
Therefore,
Again,
Since
Thus,
In the second figure, the length of the major axis greater than the length of the minor axis.
Therefore,
Now
By squaring in both sides,
Dividing both sides by
Since
Therefore,
In the third figure, the length of the major axis is little less than the length of the minor axis. In this case both the axis is almost equal.
Therefore,
Now
By squaring in both sides,
Dividing both sides by
Since
Therefore, e=1.
Conclusion:
Therefore the third figure has greater eccentricity.
Chapter 9 Solutions
Algebra and Trigonometry: Structure and Method, Book 2
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