Ellipse is Bounded in Plane

Theorem

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 \ \ $

$\ds \dfrac y b$

$=$

$\ds \sqrt {1 - \dfrac {x^2} {a^2} }$

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.

$\blacksquare$

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