Which of the following combinations will NOT define an ellipse?

• A. a,e
• B. a,f
• C. a,1/f
• D. e,f

Answer: (D)

The key idea is that an ellipse is fixed by its size and its shape. The size comes from the semi-major axis, and the shape from the eccentricity. When you know the semi-major axis together with any one of the other parameters that ties directly to the size (like the distance from the center to a focus or its reciprocal form), you can pin down all the remaining dimensions and you end up with a unique ellipse.

If you know the semi-major axis and eccentricity, you can compute the focal distance and the minor axis, so the ellipse is fully determined. If you know the semi-major axis and the focal distance (the distance from the center to a focus), you again have enough information to fix both size and shape, yielding a unique ellipse. Knowing the semi-major axis and the reciprocal of the focal distance carries the same idea: you recover the focal distance and thus the full set of dimensions.

However, knowing just the eccentricity and the focal distance alone does not lock the ellipse into a single position and orientation in the plane. That pair fixes shape and a scale element, but there’s still freedom regarding where the ellipse sits and which way it’s tilted. Without a parameter that fixes the location or orientation, this combination does not define one unique ellipse. Hence this pairing does not define an ellipse in the plane, which is why it’s the correct choice.