Plane and solid analytic geometry, by William F. Osgood and William C. Graustein.

THE ELLIPSE 103
ellipses. As a measure of the roundness or flatness of an
ellipse a number, called the eccentricity, has been chosen; this
number is defined as the radio c/a and is denoted by e:
(3) e =.
a
Since c is always less than a, it is seen that the eccentricity
of an ellipse is always less than unity:
e <.
In terms of a and b, e has the value:
(4) e a- b
a
All ellipses with the same eccentricity are similar, and conversely. For the shape of an ellipse depends only on b/a, the
ratio of its breadth to its length, and since from (4)
all ellipses for which the ratio b/a is the same have the same
eccentricity, and conversely.
A circle is the limiting case of an ellipse whose foci approach each other, the length 2a remaining constant. The
eccentricity approaches 0, and a circle is often spoken of as an
ellipse of eccentricity 0.
EXERCISES
1. The semi-axes of an ellipse are of lengths 3 cm. and 5 cm.
Find the distance between the foci, and the eccentricity.
Ans. 8; 4.
2. The eccentricity of an ellipse is 3 and the semi-axis
minor is 4 in. long. How long is the major axis?
3. The major axis of an ellipse is twice as great as the
minor axis. What is the eccentricity of the ellipse?
4. The major axis of an ellipse is 39 yards, and the eccentricity, -5. Find the minor axis.