document.write( "Question 985218: Find the center and the radius of the circle that passes through the points (4,4) , (1,3) and (8,-4).
\n" ); document.write( "What are the coordinates of the center?\r
\n" ); document.write( "\n" ); document.write( "Thank you. \n" ); document.write( "

Algebra.Com's Answer #606054 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!

\r\n" );
document.write( "Mathlover's way is rather difficult.  Substituting those points into \r\n" );
document.write( "\r\n" );
document.write( " and getting the system of three equations:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It's complicated but if you simplify all those and substitute, you'll\r\n" );
document.write( "get h, k, and r, as she has shown above.\r\n" );
document.write( "\r\n" );
document.write( "Here's an easier way but it's just about as long:  \r\n" );
document.write( "Draw the three points and connect two pairs\r\n" );
document.write( "of them.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We find the equations of the perpendicular bisectors of each chord.\r\n" );
document.write( "\r\n" );
document.write( "The slope of the shorter chord, using the slope formula is 1/3\r\n" );
document.write( "\r\n" );
document.write( "The midpoint of the shorter chord, using the midpoint formula is (5/2,7/2),\r\n" );
document.write( "\r\n" );
document.write( "So the perpendicular bisector of the short chord has slope which is the negative\r\n" );
document.write( "reciprocal of 1/3 which is -3, and it goes through (5/2,7/2)\r\n" );
document.write( "\r\n" );
document.write( "So, using the point-slope equation of a line, and simplifying, the perpendicular\r\n" );
document.write( "bisector of the shorter chord has equation y = -3x+11\r\n" );
document.write( "\r\n" );
document.write( "Doing the exact same thing with the longer chord, we find that its slope is -2\r\n" );
document.write( "and its midpoint is (6,0).  \r\n" );
document.write( "\r\n" );
document.write( "So the perpendicular bisector of the longer chord has slope which is the\r\n" );
document.write( "negative reciprocal of -2 which is 1/2, and it goes through (6,0).\r\n" );
document.write( "\r\n" );
document.write( "Using the point-slope equation of a line, and simplifying, the perpendicular\r\n" );
document.write( "bisector of the longer chord has equation .\r\n" );
document.write( "\r\n" );
document.write( "These perpendicular bisectors of the two chords are plotted in red below:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The two perpendicular bisectors (in red) must intersect at the center of the\r\n" );
document.write( "circle, so we solve the system of equations:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "and get their point of intersection as (4,-1), which is the center of the\r\n" );
document.write( "circle.\r\n" );
document.write( "\r\n" );
document.write( "We could use the distance formula to find the radius.  However it is not\r\n" );
document.write( "necessary in this case because one of the given point (4,4), just happens\r\n" );
document.write( "to be exactly 5 units above the center (4,-1), so we know that the radius\r\n" );
document.write( "is 5.\r\n" );
document.write( "\r\n" );
document.write( "So, since h=4,   k=-1, and  r=5, the standard equation   \r\n" );
document.write( "\r\n" );
document.write( " becomes\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( ".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Edwin

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

\n" );