Question

Find the standard form of the equation of the ellipse with the given characteristics?

Foci: (0, 0) and (4, 0)
Major Axis of length 8

Answer 1

Equation is `(x-2)^2/16+y^2/12=1`

Explanation:

As focii are `(0,0)` and `(4,0)`, center of ellipse is midpoint i.e. `(2,0)` and major axis is `8`, equation is of the form

`(x-2)^2/4^2+(y-0)^2/b^2=1`

where `b` is half minor axis.

As distance between focii is `4` and major axis is `8`, eccentricity is `4/8=1/2` and

`(1/2)^2=1-b^2/4^2`

or `b^2/16=1-1/4=3/4`

and `b^2=12`

Hence equation of ellipse is
`(x-2)^2/16+(y-0)^2/12=1`

or `(x-2)^2/16+y^2/12=1`

Answer 2

Equation is `(x-2)^2/16+y^2/12=1`

Explanation:

As focii are `(0,0)` and `(4,0)`, center of ellipse is midpoint i.e. `(2,0)` and major axis is `8`, equation is of the form

`(x-2)^2/4^2+(y-0)^2/b^2=1`

where `b` is half minor axis.

As distance between focii is `4` and major axis is `8`, eccentricity is `4/8=1/2` and

`(1/2)^2=1-b^2/4^2`

or `b^2/16=1-1/4=3/4`

and `b^2=12`

Hence equation of ellipse is
`(x-2)^2/16+(y-0)^2/12=1`

or `(x-2)^2/16+y^2/12=1`