An ellipse has a Center C = (0,0) and foci are located at (0,8) and (0,-8).  The length of the major axis is 24.  Fill in the missing denominators for the equation of the ellipse.          ...

An ellipse has a Center C = (0,0) and foci are located at (0,8) and (0,-8).  The length of the major axis is 24.  Fill in the missing denominators for the equation of the ellipse.

x^2 + y^2 = 1

llltkl | College Teacher | (Level 3) Valedictorian

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The foci have different y-coordinates which  indicates that  the foci lie in the y axis. Therefore, the major axis is the y axis. The equation of the ellipse has the form

`(x-h)^2/b^2+(y-k)^2/a^2=1` where center is (h, k)  and  a ≥ b > 0.

For a vertical ellipse, foci are (h, k±c) where c is the focal distance given by `sqrt(a^2-b^2)` .

Here, center =(0,0). So, from the coordinates of the foci, c=8.

From the length of the major axis, we obtain a=24/2=12. Hence

`a^2` =144.

Also, `b^2=a^2-c^2` =144-64=80

Now plugging in the values of `a^2` and `b^2` to the equation we get:

`x^2/80+y^2/144=1.`

Therefore, the required equation of the ellipse is `x^2/80+y^2/144=1.`

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