# Ellipse

An ellipse is a set of points in the plane, where the sum of the distances of the points from two fixed points is constant. The two points are called focal points.

$${r}_{1}+{r}_{2}=2a=const$$

$$e=\sqrt{{a}^{2}-{b}^{2}}$$

## Ellipse

$$\frac{(x-u{)}^{2}}{{a}^{2}}+\frac{(y-v{)}^{2}}{{b}^{2}}=1$$

## Central Ellipse

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

## Focal points of a Central Ellipse

Horizontal ellipse (a>b):

$$a>b\to {F}_{1}(-e,0);{F}_{2}(e,0);e=\sqrt{{a}^{2}-{b}^{2}}$$

Vertical ellipse (a<b):

$$a<b\to {F}_{1}(0,e);{F}_{2}(0,-e);e=\sqrt{{b}^{2}-{a}^{2}}$$

## Tangency condition of a central ellipse and a line

$${a}^{2}\xb7{k}^{2}+{b}^{2}={n}^{2}$$

where the equation of the tangent line is :

$$y=kx+n$$

Keywords: ellipse, central ellipse, equation of an ellipse, tangent to a ellipse