Eccentricity of Ellipse in terms of Semi-Major and Semi-Minor Axes

Theorem

Let $K$ be a ellipse such that:

$a$ denotes the length of the semi-major axis of $K$

$b$ denotes the length of the semi-minor axis of $K$

$e$ denotes the eccentricity of $K$.

Then:

$e^2 = 1 - \dfrac {b^2} {a^2}$

Proof

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

\(\ds a^2\)

\(=\)

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

Focus of Ellipse from Major and Minor Axis

\(\ds \leadsto \ \ \)

\(\ds c^2\)

\(=\)

\(\ds a^2 - b^2\)

\(\ds e\)

\(=\)

\(\ds \frac c a\)

Eccentricity of Ellipse is Interfocal Distance over Major Axis

\(\ds e^2\)

\(=\)

\(\ds \frac {c^2} {a^2}\)

\(\ds \)

\(=\)

\(\ds \frac {a^2 - b^2} {a^2}\)

\(\ds \)

\(=\)

\(\ds 1 - \frac {b^2} {a^2}\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ellipse
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ellipse