Circle is Ellipse with Equal Major and Minor Axes

Theorem

Let $E$ be an ellipse whose major axis is equal to its minor axis.

Then $E$ is a circle.

Proof

Let $E$ be embedded in a Cartesian plane in reduced form.

Then from Equation of Ellipse in Reduced Form $E$ can be expressed using the equation:

$\dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

where the major axis and minor axis are $a$ and $b$ respectively.

Let $a = b$.

Then:

\(\ds \dfrac {x^2} {a^2} + \dfrac {y^2} {a^2}\)

\(=\)

\(\ds 1\)

\(\ds \leadsto \ \ \)

\(\ds x^2 + y^2\)

\(=\)

\(\ds a^2\)

which by Equation of Circle center Origin is the equation of a circle whose radius is $a$.

$\blacksquare$

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {IV}$. The Ellipse: $2$. To find the equation of the ellipse in its simplest form
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): circle
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): circle