#geometry 

An **ellipse** is the set of all points in the plane the sum of whose distances from two fixed points $\LARGE F_1$ and $\LARGE F_2$ (the **foci**) is constant.

## Standard equation
The standard equation of an ellipse with foci on the x-axis and center at the origin:
$$\LARGE \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$Derivation:
![[Pasted image 20240930235831.png]]
![[Pasted image 20240930235859.png]]

![[Pasted image 20241001000928.png]]
$\LARGE 2a$ is the sum of distances from the foci to a point.
НЕЛЬЗЯ отождествлять конкретные координаты фокусов со значением $\LARGE c$, это только  в случае, если центр симметрии эллипса находится в начале координат
## Eccentricity of an ellipse
![[Pasted image 20241001001601.png]]

![[Pasted image 20241001001624.png]]

## Shifted ellipse
![[Pasted image 20241001004424.png]]

General rule for [[Shifted conics|shifting graphs of conic sections]]:
![[Pasted image 20241001004522.png]]

Ellipse directrixes:
![[Pasted image 20241112214314.png]]