document.write( "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. \n" ); document.write( "

Algebra.Com's Answer #416971 by solver91311(24713) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "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

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "An ellipse with center at

, and semi-axes of

and

is:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "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:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solve for

and

to find the semi-axes and complete the equation. I used elimination.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "John
\n" ); document.write( "

\n" ); document.write( "My calculator said it, I believe it, that settles it
\n" ); document.write( "

$\"The$

\n" ); document.write( " \n" ); document.write( "

\n" );