Question

Give standard equation for the ellipse with the given characteristics: Major axis of length 24 Foci:(plus/minus 5,0)?

Answer 1

`x^2/12^2+y^2/(sqrt119)^2=1`

Explanation:

The foci at `(-5,0) and (5,0)` tells us that the major axis is horizontally oriented, therefore, the general form is:

`(x-h)^2/a^2+(y-k)^2/b^2=1, a > b" [1]"`

The location of the foci allows us to write the following equations:

`k = 0" [2]"`
`h-sqrt(a^2-b^2) = -5" [3]"`
`h + sqrt(a^2-b^2) = 5" [4]"`

Add equation [3] to equation [4] and solve for the value of h:

`2h = 0`

`h = 0" [5]"`

Substitute equations [2] and [5] into equation [1]:

`(x)^2/a^2+(y)^2/b^2=1" [1.1]"`

The major axis length of `24` allows us to find the value of a:

`a = 24/2`

`a = 12" [6]"`:

Substitute equation [6] into equation [1.1]:

`(x)^2/12^2+(y)^2/b^2=1" [1.2]"`

Substitute equations [5] and [6] into equation [4] and solve for the value of b:

`0 + sqrt(12^2-b^2) = 5`

`-b^2 = 25-144`

`b = sqrt119" [7]"`

Substitute equation [7] into equation [1.2]:

`x^2/12^2+y^2/(sqrt119)^2=1`