Ellipses and Hyperbolas

Ellipse Standard Form

Write the standard equation of a centered ellipse. Identify its center, vertices, and co-vertices.

Example

Use a and b to write and plot a centered ellipse.

highlighted = computed this step

Step 1 — Read a and b

Read the semi-major and semi-minor lengths.

a=5b=3a= \hl{5} \quad b= \hl{3}

Step 2 — Square the axes

Square a and b for the denominators.

a2=25b2=9a^{2} = \hl{25} \quad b^{2} = \hl{9}

Step 3 — Standard form

Write the centered ellipse standard form.

x225+y29=1\frac{x^{2}}{ \hl{25} } + \frac{y^{2}}{ \hl{9} } = 1

Step 4 — Plot the ellipse

Plot the ellipse and its vertices.

Coordinate plot\text{Coordinate plot}
Ellipse standard form plotCentered ellipse with left and right vertices and upper and lower co-vertices.LRDU

Step 5 — Vertices

State the left and right vertices.

L=(-5,0)R=(5,0)L=( \hl{-5} , \hl{0} ) \quad R=( \hl{5} , \hl{0} )

Step 6 — Co-vertices

State the lower and upper co-vertices.

D=(0,-3)U=(0,3)D=( \hl{0} , \hl{-3} ) \quad U=( \hl{0} , \hl{3} )
ellipse-standard-form For a centered horizontal ellipse, x^2/a^2 + y^2/b^2 = 1. The vertices use a, and the co-vertices use b.