Question

How do you write an equation of an ellipse for the given Foci (0,±8) Co-Vertices (±8,0)?

Answer 1

`x^2/64+y^2/128=1`

Explanation:

From the given:
Foci: `F_1(0, 8), F_2(0, -8)`
Co-vertices at `(8, 0)` and (-8, 0)#

by inspection, the center of the ellipse is at Origin `(0, 0)`
that means

Center `(h, k)=(0, 0)`

Also, by inspection, `c=8` the distance from the center to a focus.
Also, `b=8` the distance from the center to one end point of the minor axis also called semi-minor axis length.

Compute for semi-major axis length `a`:

`a^2=b^2+c^2`

`a=sqrt(b^2+c^2)`

`a=sqrt(8^2+8^2)=sqrt(64+64`

`a=sqrt(2*64)`

`a=8sqrt(2)`

Use now the standard Form of the equation of ellipse for Vertical Major Axis.

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

Let us put on the values of `h=0, k=0, a=8sqrt(2), b=8`

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

`x^2/64+y^2/128=1`

Kindly check the graph ....
graph{(x^2/64+y^2/128-1)=0[-25,25,-15,15]}

Have a nice day !!! from the Philippines...