SOLUTION: Find an equation an ellipse with foci (1,-1) and (1,-7) with major axis length 10

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Question 1101761: Find an equation an ellipse with foci (1,-1) and (1,-7) with major axis length 10
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


The coordinates of the two foci are (1,-7) and (1,-1). That tells us...
(1) The center of the ellipse is (1,-4);
(2) the major axis is vertical; and
(3) the distance from the center to each focus is 3.

The standard form for the equation of an ellipse with a vertical major axis is

%28%28x-h%29%5E2%2Fb%5E2%29+%2B+%28%28y-k%29%5E2%2Fa%5E2%29+=+1

(h,k) is the center of the ellipse ((1,-4) in this example);
a is the semi-major axis (5 in this example);
b is the semi-minor axis;
and the values of a and b are related by the equation a%5E2-b%5E2+=+c%5E2, where c is the distance from the center of the ellipse to each focus (3 in this example).

In this example, since a is 5 and c is 3, b is 4.

And now we have all the specific values needed to write the equation of this ellipse:

%28%28x-1%29%5E2%2F4%5E2%29+%2B+%28%28y%2B4%29%5E2%2F5%5E2%29+=+1

or

%28%28x-1%29%5E2%2F16%29+%2B+%28%28y%2B4%29%5E2%2F25%29+=+1