# Circle is Ellipse with Equal Major and Minor Axes

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## 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