Editing Ellipse (section)

=== Standard equation ===
The standard form of an ellipse in Cartesian coordinates assumes that the origin is the center of the ellipse, the ''x''-axis is the major axis, and:
{{unbulleted list | style = padding-left:1.5em
| the foci are the points <math>F_1 = (c,\, 0),\ F_2=(-c,\, 0)</math>,
| the vertices are <math>V_1 = (a,\, 0),\ V_2 = (-a,\, 0)</math>.
}}
For an arbitrary point <math>(x,y)</math> the distance to the focus <math>(c,0)</math> is <math display="inline">\sqrt{(x - c)^2 + y^2 }</math> and to the other focus <math display="inline">\sqrt{(x + c)^2 + y^2}</math>. Hence the point <math>(x,\, y)</math> is on the ellipse whenever:
<math display="block">\sqrt{(x - c)^2 + y^2} + \sqrt{(x + c)^2 + y^2} = 2a\ .</math>

Removing the [[radical expression|radicals]] by suitable squarings and using <math>b^2 = a^2-c^2</math> (see diagram) produces the standard equation of the ellipse:<ref name="mathworld">{{cite web | url=http://mathworld.wolfram.com/Ellipse.html |title=Ellipse - from Wolfram MathWorld |publisher=Mathworld.wolfram.com |date=2020-09-10 |access-date=2020-09-10}}</ref>
<math display="block">\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,</math>
or, solved for ''y'':
<math display="block">y = \pm\frac{b}{a}\sqrt{a^2 - x^2} = \pm \sqrt{\left(a^2 - x^2\right)\left(1 - e^2\right)}.</math>

The width and height parameters <math>a,\; b</math> are called the [[semi-major and semi-minor axes]]. The top and bottom points <math>V_3 = (0,\, b),\; V_4 = (0,\, -b)</math> are the ''co-vertices''. The distances from a point <math>(x,\, y)</math> on the ellipse to the left and right foci are <math>a + ex</math> and <math>a - ex</math>.

It follows from the equation that the ellipse is ''symmetric'' with respect to the coordinate axes and hence with respect to the origin.