Answer:
The ratio (center to focus)/(semi-major axis) is the greatest
Explanation:
Given various foci coordinates and major axis lengths, you are asked to identify the ellipse with the highest eccentricity.
Ellipse Eccentricity
The eccentricity of an ellipse can be defined as the ratio of distance between foci to the length of the major axis. A high value of eccentricity corresponds to a longer, narrower ellipse.
The attached graph plots the locations of the given foci. They are color coded to correspond with the color of the resulting ellipse. (We have used the given r-value as the semi-major axis, not the major axis.)
The blue focal points (A, A') of the blue ellipse give the "flattest" of the bunch, so corresponds to the highest eccentricity. Its parameters are those of the first answer choice.
Values
If we assume the given r-value is the semi-major axis, the eccentricity represented by each of the answer choices is ...
- (15√2)/22 ≈ 0.964 . . . . highest
- (10√2)/20 ≈ 0.707
- (5√13)/22 ≈ 0.819
- (5√2)/12 ≈ 0.589
The numerator in each case is the distance from the center (0, 0) to one focus.