12.1 The ellipse  (Page 3/16)

Standard forms of the equation of an ellipse with center (0,0)

The standard form of the equation of an ellipse with center $\left(0,0\right)$ and major axis on the x-axis is

where

• $a>b$
• the length of the major axis is $2a$
• the coordinates of the vertices are $\left(\pm a,0\right)$
• the length of the minor axis is $2b$
• the coordinates of the co-vertices are $\left(\mathrm{0,}\pm b\right)$
• the coordinates of the foci are $\left(\pm c,0\right)$ , where $c^2=a^2-b^2.$ See [link] a

The standard form of the equation of an ellipse with center $\left(0,0\right)$ and major axis on the y-axis is

where

• $a>b$
• the length of the major axis is $2a$
• the coordinates of the vertices are $\left(\mathrm{0,}\pm a\right)$
• the length of the minor axis is $2b$
• the coordinates of the co-vertices are $\left(\pm b,0\right)$
• the coordinates of the foci are $\left(\mathrm{0,}\pm c\right)$ , where $c^2=a^2-b^2.$ See [link] b

Note that the vertices, co-vertices, and foci are related by the equation $c^2=a^2-b^2.$ When we are given the coordinates of the foci and vertices of an ellipse, we can use this relationship to find the equation of the ellipse in standard form.