Why is the equation of an ellipse (x/a)^2 + (y/b)^2 = 1? I've seen many proofs online, but I can't really wrap my mind around it. 
Being a generalization of the circle, I thought its equation would be as easy to understand as the circle's. Turns out I was wrong, or maybe I'm just too stupid to grasp the geometric intuition behind it. 
 A: My geometric intuition, hopefully it works for you, is that if you have $x^2+y^2=1$ then you have a circle, but if you do x/a or y/b then you end up contracting or expanding the circle in the x or in the y direction. If you have the original circle, it intersects the x axis at 1 and -1, but in the new ellipse it intersects the x axis when x/a is 1 or -1, namely a and -a. Hopefully this helps you graph them too. 
A: It is the equation of the ellipse centered at the origin.
Consider the ellipse centered at the origin and whose focal axis coincides with the axis X. The focus $F$ and $F'$ are on the x axis. $O$ as its center is a midpoint of FF' segment , the coordinates of F and F ' will for example $( c , 0 )$ and $( -c , 0 )$ , respectively , where c is a positive constant. Let p $( x , y)$ any point of the ellipse. By the definition of the curve, the point P must satisfy the condition.
$$d(FP)+d(F´P)=2a$$
where $a$ is a positive constant greater than $c$, then 
$$d(FP)= \sqrt{(x-c)^2+y^2}$$
$$d(F´P)= \sqrt{(x+c)^2+y^2}$$
and remplace.
$$\sqrt{(x-c)^2+y^2}+\sqrt{(x+c)^2+y^2}=2a$$
simplifying it.
$$cx+a^2=a*\sqrt{(x+c)^2+y^2}$$, then it square
$$c^2x^2+2a^2cx+a^4=a^2x^2+2a^2cx+a^2c^2+a^2y^2$$
$$(a^2-c^2)x^2+a^2y^2=a^2(a^2-c^2)$$
how $2a>2c$ and $a^2>c^2$ then $a^2-c^2$ is positive and get $b^2=a^2-c^2$ then
$$b^2x^2+a^2y^2=a^2b^2$$
that finally it divide for $a^2b^2$
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
