# Focus of Ellipse from Major and Minor Axis

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## 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

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:

\(\ds d\) | \(=\) | \(\ds F_1 V_2 + F_2 V_2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {a + c} + \paren {a - c}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2 a\) |

It also holds true for $C_2$:

- $F_1 C_2 + F_2 C_2 = d$

Then:

\(\ds F_1 C_2^2\) | \(=\) | \(\ds O F_1^2 + O C_2^2\) | Pythagoras's Theorem | |||||||||||

\(\ds \) | \(=\) | \(\ds c^2 + b^2\) |

and:

\(\ds F_1 C_2^2\) | \(=\) | \(\ds O F_1^2 + O C_2^2\) | Pythagoras's Theorem | |||||||||||

\(\ds \) | \(=\) | \(\ds c^2 + b^2\) |

Thus:

\(\ds F_1 C_2 + F_2 C_2\) | \(=\) | \(\ds 2 \sqrt {b^2 + c^2}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 2 a\) | as $2 a = d$ | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds \sqrt {b^2 + c^2}\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds b^2 + c^2\) | \(=\) | \(\ds a^2\) |

$\blacksquare$

## Also presented as

This result is also seen presented as:

- $c^2 = a^2 - b^2$