Firstly an ellipse is a squashed circle, and a circle is just a special type of ellipse. The general formula for an ellipse is:
(1) 
Where 
is the radius of the ellipse, 
is the radius of the ellipse in the x-direction and b is the radius of the ellipse in the y-direction. Therefore a circle of radius 1 has 
, so the formula is:
(2) 
Which looks like this:
An ellipse has the following measurements:
We also have two important points along the x-axis of the ellipse called the focus points or foci, they are usually labelled 
and 
, the x-coordinate of the foci is found using:
(3) 
where 
is the eccentricity of the ellipse, which is a measure of how squashed the ellipse is compared to a circle. The eccentricity is calculated using:
(4) 
From this we can see that the eccentricity of a circle is zero, and the most squashed ellipse will have an eccentricity of nearly 1. The range of the eccentricity of an ellipse is 
.
We now have an ellipse with the following important marks and distances:
Interestingly, the distance between 
and or is also , so:
We can use Pythagoras to help prove this. Remembering that Pythagoras states that, for a right angled triangle we have:
(5) 
So combining equations (4) and (3) into equation (5) and doing some algebra we get the following:
![\[ \begin{split} \text{Hypotenuse} & = \sqrt{F_1^2 + b^2} \\ & = \sqrt{(ae)^2 + b^2} \\ & = \sqrt{\left( a \sqrt{ 1 - \frac{b^2}{a^2}} \right)^2 + b^2} \\ & = \sqrt{\left( \sqrt{ a^2 - b^2} \right)^2 + b^2} \\ & = \sqrt{( a^2 - b^2 ) + b^2} \\ & = \sqrt{a^2} \\ & = a \end{split} \]](https://andy.bettger.net/wp-content/ql-cache/quicklatex.com-56610921a6941ef108c5c56b7f9c9e1d_l3.svg "Rendered by QuickLaTeX.com")
So we can see that the distance between and is as predicted.
