We plot the line 2x-y=3 by finding its y-intercept (0,-3), and we\r\n" );
document.write( "know that it must go through (2,1), and we sketch the circle approximately: \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( " The circle has the equation:\r\n" );
document.write( " \r\n" );
document.write( " (x-h)²+(y-k)² = r²\r\n" );
document.write( " \r\n" );
document.write( "with center (h,k) and radius r, but since the center is on the \r\n" );
document.write( "x-axis, we know that k=0, so its equation is:\r\n" );
document.write( " \r\n" );
document.write( "(x-h)²+y² = r²\r\n" );
document.write( " \r\n" );
document.write( "We know that a radius drawn to the point of tangency is \r\n" );
document.write( "perpendicular to a tangent line, so we find the equation \r\n" );
document.write( "of the line perpendicular to the give line at the point\r\n" );
document.write( "of tangency, (2,1). The green line drawn below:\r\n" );
document.write( " \r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "First we have to find the slope of the given line\r\n" );
document.write( "\r\n" );
document.write( "2x - y = 3\r\n" );
document.write( "\r\n" );
document.write( "We put it in slope-intercept form to find the slope:\r\n" );
document.write( "\r\n" );
document.write( "2x - y = 3\r\n" );
document.write( " -y = -2x + 3\r\n" );
document.write( " y = 2x - 3\r\n" );
document.write( "\r\n" );
document.write( "Compare that to \r\n" );
document.write( " \r\n" );
document.write( " y = mx + b\r\n" );
document.write( "\r\n" );
document.write( "And we see that the slope of the given tangent line is 2\r\n" );
document.write( "Therefore the slope of the green line has a slope which is\r\n" );
document.write( "the \"negative reciprocal\" of 2, which is 
. Now\r\n" );
document.write( "we use the point-slope form to find the equation of the\r\n" );
document.write( "green line:\r\n" );
document.write( "\r\n" );
document.write( "y - y1 = m(x - x1)\r\n" );
document.write( "y - 1 = (x - 2)\r\n" );
document.write( "y - 1 = x + 1 \r\n" );
document.write( " y = x + 2\r\n" );
document.write( "\r\n" );
document.write( "That's the equation of the green line, so if we find its\r\n" );
document.write( "x-intercept, that will be the center of the circle. So\r\n" );
document.write( "we set y=0 to find the x-intercept:\r\n" );
document.write( "\r\n" );
document.write( " 0 = x + 2\r\n" );
document.write( "Multiply through by 2\r\n" );
document.write( "\r\n" );
document.write( " 0 = -x + 4\r\n" );
document.write( " x = 4\r\n" );
document.write( "So now we know that the x-coordinate of the center of the\r\n" );
document.write( "circle is 4 so the center is (h,k) = (4,0) and the equation of\r\n" );
document.write( "the circle is\r\n" );
document.write( "\r\n" );
document.write( "(x-4)²+y² = r²\r\n" );
document.write( "\r\n" );
document.write( "But we still do not know the radius of the circle. We can\r\n" );
document.write( "either find the distance from (4,0) to (2,1) using the\r\n" );
document.write( "distance formula. But there is an easier way. We can just\r\n" );
document.write( "substitute the point (2,1) into the equation of the circle:\r\n" );
document.write( "\r\n" );
document.write( "(x-4)²+y² = r²\r\n" );
document.write( "(2-4)²+1² = r²\r\n" );
document.write( " (-2)²+1 = r²\r\n" );
document.write( " 4+1 = r²\r\n" );
document.write( " 5 = r²\r\n" );
document.write( "\r\n" );
document.write( "We can find the radius as 
, but all we want is\r\n" );
document.write( "the equation, and we need r² for that. So the answer is:\r\n" );
document.write( "\r\n" );
document.write( "(x-4)²+y² = r²\r\n" );
document.write( "(x-4)²+y² = 5\r\n" );
document.write( "\r\n" );
document.write( "Now we can show the complete graph:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
