SOLUTION: Write the standard equation of an ellipse with the first focus being (-4,0) and the second one being (4,0), passing through (4,1)

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Question 1119425: Write the standard equation of an ellipse with the first focus being (-4,0) and the second one being (4,0), passing through (4,1)
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


With foci at (-4,0) and (4,0), the center of the ellipse is at (0,0). The equation is of the form

x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2+=+1

The distance from the center of the ellipse to each focus is c, where c%5E2+=+a%5E2-b%5E2

Since that distance is 4, we have

a%5E2-b%5E2+=+c%5E2+=+16
a%5E2+=+b%5E2%2B16

Then the equation of the ellipse can be written as

x%5E2%2F%28b%5E2%2B16%29+%2B+y%5E2%2Fb%5E2+=+1

We can find the value of b^2 by plugging in the coordinates of the known point on the ellipse, (4,1).

16%2F%28b%5E2%2B16%29%2B1%2Fb%5E2+=+1
16%28b%5E2%29%2B%28b%5E2%2B16%29+=+b%5E2%28b%5E2%2B16%29
17b%5E2%2B16+=+b%5E4%2B16b%5E2
b%5E4-b%5E2-16+=+0
b%5E2+=+%281%2Bsqrt%2865%29%29%2F2

Then

a%5E2+=+b%5E2%2B16+=+%281%2Bsqrt%2865%29%29%2F2%2B16+=+%2833%2Bsqrt%2865%29%29%2F2

And finally the equation is

x%5E2%2F%28%2833%2Bsqrt%2865%29%29%2F2%29%2By%5E2%2F%28%281%2Bsqrt%2865%29%29%2F2%29+=+1