# Equation of Ellipse in Reduced Form/Cartesian Frame

## 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 $K$ is:

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

## Proof

By definition, the foci $F_1$ and $F_2$ of $K$ are located at $\tuple {-c, 0}$ and $\tuple {c, 0}$ respectively.

Let the vertices of $K$ be $V_1$ and $V_2$.

By definition, these are located at $\tuple {-a, 0}$ and $\tuple {a, 0}$.

Let the covertices of $K$ be $C_1$ and $C_2$.

By definition, these are located at $\tuple {0, -b}$ and $\tuple {0, b}$.

Let $P = \tuple {x, y}$ be an arbitrary point on the locus of $K$.

From the equidistance property of $K$ we have that:

- $F_1 P + F_2 P = d$

where $d$ is a constant for this particular ellipse.

From Equidistance of Ellipse equals Major Axis:

- $d = 2 a$

Also, from Focus of Ellipse from Major and Minor Axis:

- $a^2 - c^2 = b^2$

Then:

\(\displaystyle \sqrt {\paren {x - c}^2 + y^2} + \sqrt {\paren {x + c}^2 + y^2}\) | \(=\) | \(\displaystyle d = 2 a\) | Pythagoras's Theorem | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \sqrt {\paren {x + c}^2 + y^2}\) | \(=\) | \(\displaystyle 2 a - \sqrt {\paren {x - c}^2 + y^2}\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {x + c}^2 + y^2\) | \(=\) | \(\displaystyle \paren {2 a - \sqrt {\paren {x - c}^2 + y^2} }^2\) | squaring both sides | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2 + 2 c x + c^2 + y^2\) | \(=\) | \(\displaystyle 4 a^2 - 4 a \sqrt {\paren {x - c}^2 + y^2} + \paren {x - c}^2 + y^2\) | expanding | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle x^2 + 2 c x + c^2 + y^2\) | \(=\) | \(\displaystyle 4 a^2 - 4 a \sqrt {\paren {x - c}^2 + y^2} + x^2 - 2 c x + c^2 + y^2\) | further expanding | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^2 - c x\) | \(=\) | \(\displaystyle a \sqrt {\paren {x - c}^2 + y^2}\) | gathering terms and simplifying | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \paren {a^2 - c x}^2\) | \(=\) | \(\displaystyle a^2 \paren {\paren {x - c}^2 + y^2}^2\) | squaring both sides | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle c^2 x^2 - 2 c x a^2 + a^4\) | \(=\) | \(\displaystyle a^2 x^2 - 2 c x a^2 + a^2 c^2 + a^2 y^2\) | expanding | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle c^2 x^2 + a^4\) | \(=\) | \(\displaystyle a^2 x^2 + a^2 c^2 + a^2 y^2\) | simplifying | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^4 - a^2 c^2\) | \(=\) | \(\displaystyle a^2 x^2 - c^2 x^2 + a^2 y^2\) | gathering terms | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^2 \paren {a^2 - c^2}\) | \(=\) | \(\displaystyle \paren {a^2 - c^2} x^2 + a^2 y^2\) | simplifying | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle a^2 b^2\) | \(=\) | \(\displaystyle b^2 x^2 + a^2 y^2\) | substituting $a^2 - c^2 = b^2$ from $(2)$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1\) | \(=\) | \(\displaystyle \frac {x^2} {a^2} + \frac {y^2} {b^2}\) | dividing by $a^2 b^2$ |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*: $10.22$

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