Archimedes got an ellipse shrinking a circle along one direction. Then Archimedes could deduce the area of an ellipse as a generalization of the area of a circle.

An ellipse has two axes of symmetry that we call the major and the minor axes.

An ellipse is commonly defined as the locus of points P such that the sum of the distances from P to two fixed points F1, F2 (called foci) are constant.

This two foci lies on its major axis, at equal distances from the center of the ellipse.

In this page we can see (intuitively) how Archimedes' approach and this definition of an ellipse are the same.

We might use Pythagoras' theorem to calculate the position of these two foci.

From the definition, a point P in the ellipse verify:

A circle is a special case of an ellipse (when a = b). In this case the two foci points are just the same: the center of the circle.

REFERENCES

To see the relation between this definition of an ellipse and its implicit equation (and more):Weisstein, Eric W. "Ellipse." From MathWorld--A Wolfram Web Resource. Also in Wikipedia: Proofs involving the ellipse.