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

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?

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

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

Q. The eccentricity of an ellipse is defined as e = c/a. If a = 10 and c = 6, what is the eccentricity?

Solution

Eccentricity e = c/a = 6/10 = 0.6.

Correct Answer: B — 0.8

Q. The equation of an ellipse is given by 4x^2 + 9y^2 = 36. What is the eccentricity of the ellipse?

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

Q. The foci of the ellipse x^2/25 + y^2/16 = 1 are located at which points?

Solution

For the ellipse, c = √(a^2 - b^2) = √(25 - 16) = √9 = 3. The foci are at (±3, 0).

Correct Answer: B — (±4, 0)

Q. What is the area of an ellipse with semi-major axis 7 and semi-minor axis 4?

Solution

The area of an ellipse is given by A = πab. Here, A = π * 7 * 4 = 28π.

Correct Answer: A — 28π

Q. What is the equation of an ellipse with foci at (±c, 0) and vertices at (±a, 0)?

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

Q. What is the length of the latus rectum of the ellipse x^2/36 + y^2/16 = 1?

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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