Standard Forms of Equations for Ellipses Centered at (h,k)

An ellipse centered at (h, k) and either a horizontal or a vertical major axis of length 2a satisfies one of the following equations, where a > b > 0 and LaTeX: c^2=a^2-b^2 $c^2=a^2-b^2$ with LaTeX: c>0 $c>0$ .

Horizontal Ellipse Centered at (h,k)

Horizontal Ellipse Centered at (h,k).PNG

$LaTeX: \frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$ $\frac{\left(x-h\right)^2}{a^2}+\frac{\left(y-k\right)^2}{b^2}=1$

Major axis is horizontal, with equation LaTeX: y=b. $y=b.$

Foci: $LaTeX: \left(h\pm c,k\right)$ $\left(h\pm c,k\right)$

Vertices: $LaTeX: \left(h\pm a,k\right)$ $\left(h\pm a,k\right)$ , and $LaTeX: \left(h,k\pm b\right)$ $\left(h,k\pm b\right)$

Vertical Ellipse Centered at (h,k)

Vertical Ellipse Centered at (h,k).PNG

$LaTeX: \frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$ $\frac{\left(x-h\right)^2}{b^2}+\frac{\left(y-k\right)^2}{a^2}=1$

The major axis is vertical, with equation LaTeX: x=b $x=b$ .

Foci: $LaTeX: \left(h,k\pm c\right)$ $\left(h,k\pm c\right)$

Vertices: $LaTeX: \left(h,k\pm a\right)$ $\left(h,k\pm a\right)$ , and $LaTeX: \left(h\pm b,k\right)$ $\left(h\pm b,k\right)$