Ellipse
Q. If an ellipse has a semi-major axis of 10 and a semi-minor axis of 6, what is the value of b^2?
Solution
For an ellipse, b is the semi-minor axis. Here, b = 6, so b^2 = 6^2 = 36.
Correct Answer: A — 36
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Q. If the equation of an ellipse is 9x^2 + 16y^2 = 144, what are the lengths of the semi-major and semi-minor axes?
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A.
3, 4
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B.
4, 3
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C.
6, 8
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D.
8, 6
Solution
Rewriting the equation in standard form gives (x^2/16) + (y^2/9) = 1, so semi-major axis a = 4 and semi-minor axis b = 3.
Correct Answer: A — 3, 4
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Q. If the lengths of the semi-major and semi-minor axes of an ellipse are 5 and 3 respectively, what is the distance between the foci?
Solution
The distance between the foci is given by 2c, where c = √(a^2 - b^2). Here, c = √(5^2 - 3^2) = √16 = 4, so the distance is 2c = 8.
Correct Answer: A — 4
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Q. The eccentricity of an ellipse is defined as e = c/a. If a = 10 and c = 6, what is the eccentricity?
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A.
0.6
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B.
0.8
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C.
0.4
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D.
0.5
Solution
Eccentricity e = c/a = 6/10 = 0.6.
Correct Answer: B — 0.8
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Q. The equation of an ellipse is given by 4x^2 + 9y^2 = 36. What is the eccentricity of the ellipse?
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A.
0.5
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B.
0.6
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C.
0.7
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D.
0.8
Solution
Rewriting gives x^2/9 + y^2/4 = 1. Here, a^2 = 9, b^2 = 4, c = √(a^2 - b^2) = √(9 - 4) = √5. Eccentricity e = c/a = √5/3 ≈ 0.6.
Correct Answer: B — 0.6
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Q. The foci of the ellipse x^2/25 + y^2/16 = 1 are located at which points?
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A.
(±3, 0)
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B.
(±4, 0)
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C.
(±5, 0)
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D.
(±6, 0)
Solution
For the ellipse, c = √(a^2 - b^2) = √(25 - 16) = √9 = 3. The foci are at (±3, 0).
Correct Answer: B — (±4, 0)
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Q. What is the area of an ellipse with semi-major axis 7 and semi-minor axis 4?
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A.
28π
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B.
14π
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C.
21π
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D.
35π
Solution
The area of an ellipse is given by A = πab. Here, A = π * 7 * 4 = 28π.
Correct Answer: A — 28π
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Q. What is the equation of an ellipse with foci at (±c, 0) and vertices at (±a, 0)?
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A.
x^2/a^2 + y^2/b^2 = 1
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B.
y^2/a^2 + x^2/b^2 = 1
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C.
x^2/b^2 + y^2/a^2 = 1
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D.
y^2/b^2 + x^2/a^2 = 1
Solution
The standard form of the equation of an ellipse with horizontal major axis is x^2/a^2 + y^2/b^2 = 1.
Correct Answer: A — x^2/a^2 + y^2/b^2 = 1
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Q. What is the length of the latus rectum of the ellipse x^2/36 + y^2/16 = 1?
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A.
8/3
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B.
12
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C.
16/3
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D.
24
Solution
The length of the latus rectum is given by (2b^2/a). Here, b^2 = 16, a^2 = 36, so length = (2*16/6) = 12.
Correct Answer: B — 12
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