Focus of Ellipse from Major and Minor Axis

Theorem
Let $K$ be an ellipse whose major axis is $2 a$ and whose minor axis is $2 b$.

Let $c$ be the distance of the foci of $K$ from the center.

Then:
 * $a^2 = b^2 + c^2$

Proof

 * EllipseFocus MajorMinorAxes.png

Let the foci of $K$ be $F_1$ and $F_2$.

Let the vertices of $K$ be $V_1$ and $V_2$.

Let the covertices of $K$ be $C_1$ and $C_2$.

Let $P = \tuple {x, y}$ be an arbitrary point on the locus of $K$.

From the equidistance property of $K$ we have that:


 * $F_1 P + F_2 P = d$

where $d$ is a constant for this particular ellipse.

This is true for all points on $K$.

In particular, it holds true for $V_2$, for example.

Thus:

It also holds true for $C_2$:


 * $F_1 C_2 + F_2 C_2 = d$

Then:

and:

Thus:

Also presented as
This result is also seen presented as:
 * $c^2 = a^2 - b^2$