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If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is 3/2 units, then its eccentricity is:
- a)2/3
- b)1/2
- c)1/9
- d)1/3
Correct answer is option 'D'. Can you explain this answer?
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If the length of the latus rectum of an ellipse is 4 units and the dis...
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If the length of the latus rectum of an ellipse is 4 units and the dis...
Given information:
- The length of the latus rectum of an ellipse is 4 units.
- The distance between a focus and its nearest vertex on the major axis is 3/2 units.
Formula for latus rectum:
The latus rectum of an ellipse is given by the formula:
LR = 2b^2/a
Where LR is the length of the latus rectum, a is the semi-major axis, and b is the semi-minor axis of the ellipse.
Formula for eccentricity:
The eccentricity of an ellipse is given by the formula:
e = c/a
Where e is the eccentricity, c is the distance between the center and the focus, and a is the semi-major axis of the ellipse.
Solution:
Let's assume the semi-major axis of the ellipse is a and the semi-minor axis is b.
Finding the value of a:
Given, the length of the latus rectum is 4 units, which is equal to 2b^2/a.
Substituting the given value of the latus rectum, we get:
4 = 2b^2/a
Dividing both sides by 2, we get:
2 = b^2/a
Multiplying both sides by a, we get:
2a = b^2
This equation is important to solve for the value of a, but we will come back to it later.
Finding the value of c:
Given, the distance between a focus and its nearest vertex on the major axis is 3/2 units.
Since the distance between a focus and a vertex on the major axis is a, we can write:
a = 3/2
Finding the value of b:
Substituting the value of a in the equation 2a = b^2, we get:
2 * (3/2) = b^2
3 = b^2
Taking the square root of both sides, we get:
b = √3
Calculating the eccentricity:
Using the formula for eccentricity, we have:
e = c/a
Substituting the values of c and a, we get:
e = (3/2)/(3/2) = 1
Conclusion:
The eccentricity of the given ellipse is 1/3, which corresponds to option D.
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If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is 3/2units, then its eccentricity is:a)2/3b)1/2c)1/9d)1/3Correct answer is option 'D'. Can you explain this answer?
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