Relationship Between Semi-Major and Semi-Minor Axes

Look at the leftmost point on the ellipse:

a c c

The distance to the closer focal point must be

a − c

and the distance to the further focal point must be

a + c

Therefore, the sum of the distances to the focal points must be:

(a − c) + (a + c) = 2a

As, by definition, this value holds for every point on the ellipse, it must hold for the topmost point:

c c b

The two triangles formed between focal point, centre, and point on the ellipse must be right triangles and congruent. The line from the point on the ellipse to the focal point is the hypotenuse of the triangle and so the length must be a (because the two hypotenuses are equal and sum to 2a). By Pythagoras’s Theorem, we therefore have:

b2 + c2 = a2

or alternatively:

c2 = a2 − b2