For simplicity, let's say that the ellipse is horizontal, centered at $\left(0\, 0\right)$ with the following foci: E at $\left(-c\, 0\right)$ and F at $\left(c\, 0\right)$. Thus, the distance from each focus to the center is c.

The distance from a general point $P\left(x\, y\right)$ on the ellipse to E is given by $\mathrm{PE} \= \sqrt{{\left(x - \left(-c\right)\right)}^2\+{\left(y - 0\right)}^2} \= \sqrt{{\left(x\+c\right)}^2\+y^2}$.

The distance from P to F is given by $\mathrm{PF} \= \sqrt{{\left(x - c\right)}^2\+{\left(y -0\right)}^2}\= \sqrt{{\left(x\mathit{ }\mathit{-}\mathit{ }c\right)}^2\+y^2}$.

Looking at the case in which P is a vertex of the ellipse and adding up the distances from this vertex to each focus, we see that the sum of these distances is $2a$.

Thus, we know that:         

$\mathrm{PE}\+\mathrm{PF}\=2a$

$\sqrt{{\left(x\+c\right)}^2\+y^2}\+\sqrt{{\left(x-c\right)}^2\+y^2}\=2a$

$\sqrt{{\left(x\+c\right)}^2\+y^2} \= 2a - \sqrt{{\left(x-c\right)}^2\+y^2}$

${\left(x\+c\right)}^2\+y^2 \= {\left(2 a - \sqrt{{\left(x-c\right)}^2\+y^2}\right)}^2$

$x^2\+2cx\+c^2\+y^2\=4a^2-4a\sqrt{{\left(x-c\right)}^2\+y^2}\+{\left(x-c\right)}^2\+y^2$

$x^2\+2cx\+c^2\+y^2 \= 4a^2-4a\sqrt{{\left(x-c\right)}^2\+y^2} \+ x^2 - 2 c x \+ c^2 \+ y^2$

$2c x \= 4 a^2-4a\sqrt{{\left(x-c\right)}^2\+y^2} - 2 c x$

$4c x - 4 a^2 \= -4 a \sqrt{{\left(x-c\right)}^2\+y^2}$

$c x - a^2 \= -a \sqrt{{\left(x-c\right)}^2\+y^2}$

${\left(c x - a^2\right)}^2 \= a^2\left({\left(x-c\right)}^2\+y^2\right)$

$c^2{x}^2-2a^{2}cx\+ a^4 \= a^2{x}^2-2 a^{2}cx\+ a^2{c}^{2 }\+a^2{y}^2$

$c^2{x}^2\+ a^4 \= a^2{x}^2\+ a^2{c}^{2 }\+a^2{y}^2$

$a^4-a^2{c}^2\=a^2{x}^2-c^2{x}^2 \+ a^2{y}^{2 }$

$a^2\left(a^2-c^2\right)\=\left(a^2-c^2\right)x^2\+a^2{y}^2$

Considering the case when P is the co-vertex located at $\left(0\,b\right)$, we see that $b^2 \+ c^2 \= a^2$, so $b^2 \= a^2 - c^2$. Substituting $b^2$, we get:

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

$\frac{a^2{b}^2}{a^2{b}^2} \= \frac{b^2{x}^2\+ a^2{y}^2}{a^2{b}^2}$

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

This is the standard equation for an ellipse centered at $\left(0\, 0\right)$ with horizontal semi-major axis a and semi-minor axis b.