Ellipse is Bounded in Plane

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

Then $E$ is bounded.


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

Let $E$ be expressed in reduced form:

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


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.