document.write( "Question 812802: The vertex and focus of the parabola y^2-4x+20=0 are one focus and one vertex respectively of an ellipse whose center is the origin. Find the equation of the ellipse \n" ); document.write( "

Algebra.Com's Answer #489343 by Edwin McCravy(20059) $\"\"$ $\"About$

You can put this solution on YOUR website!
The vertex and focus of the parabola y^2-4x+20=0 are one focus and one vertex respectively of an ellipse whose center is the origin. Find the equation of the ellipse
\n" ); document.write( "

\r\n" );
document.write( "y²-4x+20 = 0  (y is squared so it is a parabola \r\n" );
document.write( "               with horizontal axis of symmetry)\r\n" );
document.write( "\r\n" );
document.write( "      y² = 4x-20\r\n" );
document.write( "\r\n" );
document.write( "  (y-0)² = 4(x-5)\r\n" );
document.write( "\r\n" );
document.write( "Compare to\r\n" );
document.write( "\r\n" );
document.write( "  (y-k)² = 4p(x-h), with vertex (h,k),\r\n" );
document.write( "\r\n" );
document.write( "so vertex = (5,0)\r\n" );
document.write( "\r\n" );
document.write( "and since 4p=4, p=1 so distance from\r\n" );
document.write( "vertex to focus is |p| = 1.\r\n" );
document.write( "\r\n" );
document.write( "p is positive so it opens to the right.\r\n" );
document.write( "\r\n" );
document.write( "The graph of the parabola is \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So the focus is |p|=1 unit to the right \r\n" );
document.write( "of the vertex, or (6,0).\r\n" );
document.write( "\r\n" );
document.write( "Now we want the equation of an ellipse with center (0,0)\r\n" );
document.write( "focus (5,0), and vertex (6,0), like this green ellipse:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In an ellipse the distance from center to vertex is \"a\",\r\n" );
document.write( "the semi-major axis, and the distance from (0,0) to (6,0) is a=6.\r\n" );
document.write( "The distance from center to focus is \"c\", and the distance from \r\n" );
document.write( "(0,0,) to (5,0) is c=5.  The Pythagorean relationship for all \r\n" );
document.write( "ellipses is\r\n" );
document.write( "\r\n" );
document.write( "c² = a²-b² where \"b\" is the semi-minor axis. So\r\n" );
document.write( "5² = 6²-b²\r\n" );
document.write( "25 = 36-b²\r\n" );
document.write( "b² = 36-25\r\n" );
document.write( "b² = 11\r\n" );
document.write( " b = √11 \r\n" );
document.write( "\r\n" );
document.write( "This ellipse has horizontal major axis, and its equation is\r\n" );
document.write( "\r\n" );
document.write( "  1\r\n" );
document.write( " \r\n" );
document.write( "  1\r\n" );
document.write( " \r\n" );
document.write( "  1\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );