# Ellipse is Bounded in Plane

## Theorem

Let $E$ be an ellipse embedded in in a Euclidean plane.

Then $E$ is bounded.

## Proof

Let a Cartesian coordinate system be applied to the Euclidean plane in which $E$ is embedded.

Let $E$ be expressed in reduced form:

 $\ds \dfrac {x^2} {a^2} + \dfrac {y^2} {b^2}$ $=$ $\ds 1$ $\ds \leadsto \ \$ $\ds \dfrac x a$ $=$ $\ds \sqrt {1 - \dfrac {y^2} {b^2} }$ $\ds \leadsto \ \$ $\ds \dfrac y b$ $=$ $\ds \sqrt {1 - \dfrac {x^2} {a^2} }$

Hence:

there are no real values of $x$ for $\size y > b$
there are no real values of $y$ for $\size x > b$.

Hence $E$ exists entirely within the rectangle whose sides are $x = \pm a$ and $y = \pm b$.

The result follows.

$\blacksquare$