Ellipses and Hyperbolas

Ellipse Foci

Find the foci of a centered ellipse from a and b.

Example

Use c squared equals a squared minus b squared to locate foci.

highlighted = computed this step

Step 1 — Read a and b

Read a and b for the ellipse.

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

Step 2 — Compute c squared

Subtract b squared from a squared.

c2=25−9=16c^{2} = 25 - 9 = \hl{16}

Step 3 — Take the square root

Take the square root to find c.

c=4c= \hl{4}

Step 4 — Foci

Place the foci c units from the center.

F=(-4,0)G=(4,0)F=( \hl{-4} , \hl{0} ) \quad G=( \hl{4} , \hl{0} )

Step 5 — Plot the foci

Plot the ellipse and foci.

Coordinate plot\text{Coordinate plot}
Ellipse foci plotCentered ellipse with the two foci on the major axis.FG

Step 6 — Result

State c and the two foci.

c=4F=(-4,0)G=(4,0)c= \hl{4} \quad F=( \hl{-4} , \hl{0} ) \quad G=( \hl{4} , \hl{0} )
ellipse-foci For a horizontal ellipse with a > b, use c^2 = a^2 - b^2. The foci are (-c, 0) and (c, 0).