Question

Which of the following statements about the foci and the eccentricity is correct?:

(A) The foci are (0,-313) and (0,313), and the eccentricity is 13
(B) The foci are (0,-313) and (0,313), and the eccentricity is 13.
(C) The foci are (-313,0) and (313,0), and the eccentricity is ~3
(D) The foci are (-313,0) and (313,0), and the eccentricity is 1/3.

Answer 1

The correct statement is:

(C) The foci are (-313,0) and (313,0), and the eccentricity is ~3.

Step-by-step explanation:

In the context of conic sections, such as ellipses, the foci are points that help define the shape of the ellipse, and the eccentricity is a measure of how "stretched out" or elongated the ellipse is. The eccentricity (\(e\)) is a positive number less than 1 for ellipses.

For an ellipse centered at the origin with foci along the major axis, the coordinates of the foci are (\( \pm c, 0 \)), where \(c\) is the distance from the center to each focus. The eccentricity (\(e\)) is related to the distance between the foci (\(2c\)) and the length of the major axis (\(2a\)) by the formula \(e = \frac{c}{a}\).

In option (C), the foci are given as (-313,0) and (313,0), indicating that \(c = 313\). Additionally, the eccentricity is given as approximately 3. The relationship \(e = \frac{c}{a}\) implies that \(a = \frac{c}{e} \approx \frac{313}{3}\), which is consistent with the given foci and eccentricity for an ellipse.

Therefore, option (C) is the correct statement.

Answer 2

Remembering that the eccentricity of a circle is zero (since the foci coincide at the center), statement (A) and (B) suggesting an eccentricity of 13 are incorrect because no ellipse can have an eccentricity greater than 1. Statement (C), which suggests an eccentricity of about 3, is also incorrect for the same reason.

Step-by-step explanation:

To determine which statement about the foci and the eccentricity is correct, we should recall that the eccentricity (e) of an ellipse is defined as the ratio of the distance between the two foci to the length of the major axis. For an ellipse, 0 ≤ e < 1. However, the given foci being (0,-313) and (0,313) or (-313,0) and (313,0) suggest a distance between foci of 626, which corresponds to an ellipse that is very elongated along one axis. Yet, without the length of the major axis, we cannot calculate the exact eccentricity. Therefore, based on the information provided, all statements given about the eccentricity could be speculative unless additional context is presented about the length of the major axis of the ellipse.

Remembering that the eccentricity of a circle is zero (since the foci coincide at the center), statement (A) and (B) suggesting an eccentricity of 13 are incorrect because no ellipse can have an eccentricity greater than 1. Statement (C), which suggests an eccentricity of about 3, is also incorrect for the same reason. That leaves us with statement (D), stating the foci are (-313,0) and (313,0), and the eccentricity is 1/3, which is within the possible range for an ellipse. However, without the length of the major axis, this claim cannot be validated.

Hence, none of the options can be definitively stated as correct without additional information about the major axis.