Question

Question #2bd62

Answer 1

`(x-0)^2/10^2+(y-0)^2/8^2=1`

Explanation:

The standard form for the equation of an ellipse with horizontally oriented foci is:

`(x-h)^2/a^2+(y-k)^2/b^2=1; a > b`

The foci are `(h-sqrt(a^2-b^2),k)` and `(h+sqrt(a^2-b^2),k)`
The co-vertices are `(h,k-b)` and `(h,k+b)`

The co-vertices allow us to write the following equations:

`h = 0" [1]"`
`k-b = -8" [2]"`
`k + b = 8" [3]"`

To find the value of k add equation [2] to equation [3]:

`2k = 0`

`k = 0`

Substitute the value of k into equation [3], to find the value of `b`:

`0 + b = 8`

`b = 8`

Substitute this into the standard form:

`(x-0)^2/a^2+(y-0)^2/8^2=1" [4]"`

Use the x coordinate of the focus `(6,0)` to find the value of `a`:

`6 = 0+sqrt(a^2-8^2)`

`36 = a^2-64`

`a^2 = 100`

`a = 10`

Substitute the value for "a" into equation [4]:

`(x-0)^2/10^2+(y-0)^2/8^2=1`