Equation of Ellipse in Reduced Form/Cartesian Frame/Parametric Form

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Theorem

Let $K$ be an ellipse aligned in a cartesian coordinate plane in reduced form.


Let:

the major axis of $K$ have length $2 a$
the minor axis of $K$ have length $2 b$.


The equation of $K$ in parametric form is:

$x = a \cos \theta, y = b \sin \theta$


Proof

Let the point $\left({x, y}\right)$ satisfy the equations:

$x = a \cos \theta$
$y = b \sin \theta$

Then:

\(\displaystyle \frac {x^2} {a^2} + \frac {y^2} {b^2}\) \(=\) \(\displaystyle \frac {\left({a \cos \theta}\right)^2} {a^2} + \frac {\left({b \sin \theta}\right)^2} {b^2}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {a^2} {a^2} \cos^2 \theta + \frac {b^2} {b^2} \sin^2 \theta\)
\(\displaystyle \) \(=\) \(\displaystyle \cos^2 \theta + \sin^2 \theta\)
\(\displaystyle \) \(=\) \(\displaystyle 1\) Sum of Squares of Sine and Cosine

$\blacksquare$


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