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

Theorem

Let $K$ be an ellipse aligned in a cartesian 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 $\tuple {x, y}$ satisfy the equations:

$x = a \cos \theta$

$y = b \sin \theta$

Then:

\(\ds \frac {x^2} {a^2} + \frac {y^2} {b^2}\)

\(=\)

\(\ds \frac {\paren {a \cos \theta}^2} {a^2} + \frac {\paren {b \sin \theta}^2} {b^2}\)

\(\ds \)

\(=\)

\(\ds \frac {a^2} {a^2} \cos^2 \theta + \frac {b^2} {b^2} \sin^2 \theta\)

\(\ds \)

\(=\)

\(\ds \cos^2 \theta + \sin^2 \theta\)

\(\ds \)

\(=\)

\(\ds 1\)

Sum of Squares of Sine and Cosine

$\blacksquare$

Sources

• Weisstein, Eric W. "Ellipse." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Ellipse.html