SOLUTION: Find the equation of an ellipse that passes through A(3,5) and B(7,5/3) whose axis coincides with the coordinate axis.

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Question 1130622: Find the equation of an ellipse that passes through A(3,5) and B(7,5/3) whose axis coincides with the coordinate axis.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


An ellipse has two axes; and there are two coordinate axes. So when the statement of the problem sloppily says "the" axis of the ellipse coincides with "the" coordinate axis, I will assume that the intended meaning is that both axes of the ellipse coincide with both coordinate axes -- i.e., the center of the ellipse is at the origin.

Then the equation of the ellipse is

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

Form two equations in a^2 and b^2 using the two known points on the ellipse and solve the pair of equations to find the equation of the ellipse.

9%2Fa%5E2%2B25%2Fb%5E2+=+1
49%2Fa%5E2%2B%2825%2F9%29%2Fb%5E2+=+1

Multiply the second equation by 9 and subtract one equation from the other to eliminate b:

441%2Fa%5E2%2B25%2Fb%5E2+=+9
432%2Fa%5E2+=+8
8a%5E2+=+432
a%5E2+=+54

Substitute in either original equation to find b^2:

9%2F54%2B25%2Fb%5E2+=+1
25%2Fb%5E2+=+1-9%2F54+=+1-1%2F6+=+5%2F6
5b%5E2+=+150
b%5E2+=+30

We nowknow a^2 and b^2 for the ellipse, so we can write its equation:

ANSWER: x%5E2%2F54%2By%5E2%2F30+=+1