Question

4. We can compute the eccentricity of an ellipse with the equation e = c/a where a is the distance from the center of the ellipse to either vertex, and c is the distance from the center to either focus. We also know that 0<e< 1. Write a brief paragraph describing what happens to the ellipse as we change the eccentricity and let it get closer to 0, then describe what happens to the ellipse as we let the eccentricity get closer to 1.

Answer 1

For any ellipse, eccentricity e is defined as

e=c/a,

where c is the distance from the center to either focus and a is the distance from the center of ellipse to the either vertex. The term eccentricity refers to the ovalness of the shape, which basically indicates that how much the ellipse is stretched.

And we know that

0<e<1.

Now if we change the eccentricity and let it get closer to 0, that is the ratio c/a tends to zero, which means either c tends to zero or a tends to infity. Here we consider the case c tends to zero, It implies, the focus approaches to the center. Hence ovalness increases, thus the ellipse is almost in the least stretched condition.

Therefore the shape ellipse approaches to a circle.

Now if eccentricity tends to 1, which leads to the fact that distance between focus and center approches the distance between center and the either vertices. In this case ellipse is almost completely stretched and the shape approaches a straight line.