document.write( "Question 558308: Find the length of the chord of the ellipse {(x^2)/(a^2)}+{(y^2)/(b^2)}=1 directed along the diagonal of the rectangle constructed on the axes of the ellipse?
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #363019 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We need to find the distance between the two points where the green line\r\n" );
document.write( "intersects the red ellipse.\r\n" );
document.write( "\r\n" );
document.write( "First find the equation of the green line:\r\n" );
document.write( "\r\n" );
document.write( "It passes through the origin (0,0) and the point (a,b)\r\n" );
document.write( "\r\n" );
document.write( "We find its slope:\r\n" );
document.write( "\r\n" );
document.write( "m = \r\n" );
document.write( " \r\n" );
document.write( "m =  = \r\n" );
document.write( "\r\n" );
document.write( "Using the point-slope formula:\r\n" );
document.write( "\r\n" );
document.write( "y - y₁ = m(x - x₁)\r\n" );
document.write( "\r\n" );
document.write( "y - 0 = (x - 0)\r\n" );
document.write( "\r\n" );
document.write( "y = x\r\n" );
document.write( "\r\n" );
document.write( "To find the endpoints of the chord, where\r\n" );
document.write( "the green line intersects the red ellipse,\r\n" );
document.write( "we solve the system of equations:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We clear each of fractions:\r\n" );
document.write( "\r\n" );
document.write( "b²x² + a²y² = a²b²\r\n" );
document.write( "         ay = bx\r\n" );
document.write( "\r\n" );
document.write( "Square both sides of the second equation:\r\n" );
document.write( "\r\n" );
document.write( "       a²y² = b²x²\r\n" );
document.write( "\r\n" );
document.write( "Substitute b²x² for a²y² in \r\n" );
document.write( "\r\n" );
document.write( "b²x² + b²x² = a²b²\r\n" );
document.write( "\r\n" );
document.write( "      2b²x² = a²b²\r\n" );
document.write( "\r\n" );
document.write( "         x² = \r\n" );
document.write( "\r\n" );
document.write( "         x² = \r\n" );
document.write( "\r\n" );
document.write( "          x =  \r\n" );
document.write( "\r\n" );
document.write( "          x = \r\n" );
document.write( "\r\n" );
document.write( "          x =  \r\n" );
document.write( "\r\n" );
document.write( "Substitute a²y² for b²x² in \r\n" );
document.write( "\r\n" );
document.write( "a²y² + a²y² = a²b²\r\n" );
document.write( "\r\n" );
document.write( "      2a²y² = a²b²\r\n" );
document.write( "\r\n" );
document.write( "         y² = \r\n" );
document.write( "\r\n" );
document.write( "         y² = \r\n" );
document.write( "\r\n" );
document.write( "          y =  \r\n" );
document.write( "\r\n" );
document.write( "          y = \r\n" );
document.write( "\r\n" );
document.write( "          y = \r\n" );
document.write( "\r\n" );
document.write( "So the end points of the chord are\r\n" );
document.write( "\r\n" );
document.write( " and  \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We use the distance formula to find the length of the chord:\r\n" );
document.write( "\r\n" );
document.write( "d = \r\n" );
document.write( "\r\n" );
document.write( "d = \r\n" );
document.write( "\r\n" );
document.write( "d = \r\n" );
document.write( "\r\n" );
document.write( "d = \r\n" );
document.write( "\r\n" );
document.write( "d = \r\n" );
document.write( "\r\n" );
document.write( "d = \r\n" );
document.write( "\r\n" );
document.write( "That's it.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\n" ); document.write( "

\n" );