Question 670534: An ellipse is drawn to pass through the points (1, 12), (8, 10) and (1, -4), and to have the line x = 4 as an axis of symmetry. Find the coordinates of its foci and define axis of symmetry.
Answer by solver91311(24713)   (Show Source):
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Since  is an axis of symmetry, the center must lie on that vertical line, hence the  -coordinate of the center of the ellipse must be 4. Further, since the two points ) and ) are equidistant from the given axis, the center must have a  -coordinate equal to the -coordinate of the midpoint of the segment that joins those two points, namely 4. In sum, the center is at )
An ellipse with center at ) , and semi-axes of  and  is:
^2}{a^2}\ +\ \frac{(y\ -\ k)^2}{b^2}\ =\ 1)
Using the coordinates of two of the given points, and our knowledge of the location of the center, we create the following system of non-linear equations:
Solve for and to find the semi-axes and complete the equation. I used elimination.
John
My calculator said it, I believe it, that settles it
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