\r\n" );
document.write( "The line x - 2y = -4 is tangent to the circle at (0,2). the line\r\n" );
document.write( "y = 2x - 7 is tangent to the circle at (3,-1). Find the center of \r\n" );
document.write( "the circle.\r\n" );
document.write( "\r\n" );
document.write( "We will first draw the picture to see\r\n" );
document.write( "what's going on, so we'll know what to\r\n" );
document.write( "do.\r\n" );
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document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "The red line is the line x - 2y = -4.\r\n" );
document.write( "The green line is the line y = 2x - 7\r\n" );
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document.write( "There is a theorem from plane geometry that goes:\r\n" );
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document.write( "If a line is tangent to a circle, then the line is\r\n" );
document.write( "perpendicular to the radius drawn to the point of \r\n" );
document.write( "tangency.\r\n" );
document.write( "\r\n" );
document.write( "By considering the line of which this radius is a\r\n" );
document.write( "a segment, we have this corollary:\r\n" );
document.write( "\r\n" );
document.write( "If a line is tangent to a circle, then the line \r\n" );
document.write( "perpendicular to the tangent line passing through \r\n" );
document.write( "the point of tangency also passes through the center \r\n" );
document.write( "of the circle.\r\n" );
document.write( "\r\n" );
document.write( "So we find equations of perpendiculars to those two \r\n" );
document.write( "tangent lines. Both willl pass through the center,\r\n" );
document.write( "and thus their point of intersection must be the \r\n" );
document.write( "center of the circle:\r\n" );
document.write( "\r\n" );
document.write( "First we will find the equation of a line perpendicular\r\n" );
document.write( "to the red line x - 2y = -4:\r\n" );
document.write( "\r\n" );
document.write( "First we find its slope by placing it in \r\n" );
document.write( "slope-intercept form. We solve for y:\r\n" );
document.write( "\r\n" );
document.write( " x - 2y = -4\r\n" );
document.write( " -2y = -x - 4\r\n" );
document.write( " y = (1/2)x + 4\r\n" );
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document.write( "Compare that to \r\n" );
document.write( " y = mx + b\r\n" );
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document.write( "and we see that its slope is\r\n" );
document.write( "m = 1/2.\r\n" );
document.write( "\r\n" );
document.write( "A line perpendicular to\r\n" );
document.write( "it will have slope which is the\r\n" );
document.write( "reciprocal of 1/2 with its sign\r\n" );
document.write( "changed. Thus we will use m = -2/1\r\n" );
document.write( "or m = -2. We want the line\r\n" );
document.write( "to go through the point of \r\n" );
document.write( "tangency (0,2), so we use the\r\n" );
document.write( "point-slope formula:\r\n" );
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document.write( "y - y1 = m(x - x1)\r\n" );
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document.write( "with m = -2 and (x1,y1) = (0,2)\r\n" );
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document.write( "y - 2 = -2(x - 0)\r\n" );
document.write( "y - 2 = -2x\r\n" );
document.write( " y = -2x + 2\r\n" );
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document.write( "So if we draw in that perpendicular line (in brown), \r\n" );
document.write( "we have\r\n" );
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\r\n" );
document.write( "The brown line passes through the center of the circle\r\n" );
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document.write( "Now we do the same with the green\r\n" );
document.write( "line y = 2x - 7. That is, we will \r\n" );
document.write( "find the equation of a line\r\n" );
document.write( "perpendicular to the line y = 2x - 7\r\n" );
document.write( "\r\n" );
document.write( "y = 2x - 7 is already in \r\n" );
document.write( "slope-intercept form. \r\n" );
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document.write( "Comparing\r\n" );
document.write( " y = 2x - 7\r\n" );
document.write( "to \r\n" );
document.write( " y = mx + b\r\n" );
document.write( "\r\n" );
document.write( "we see that its slope is\r\n" );
document.write( "m = 2.\r\n" );
document.write( "\r\n" );
document.write( "A line perpendicular to\r\n" );
document.write( "it will have slope which is the\r\n" );
document.write( "reciprocal of 2 with its sign\r\n" );
document.write( "changed. Thus we will use \r\n" );
document.write( "m = -1/2. We want the line\r\n" );
document.write( "to go through the point of \r\n" );
document.write( "tangency (3,-1), so we use the\r\n" );
document.write( "point-slope formula again:\r\n" );
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document.write( "y - y1 = m(x - x1)\r\n" );
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document.write( "with m = -1/2 and (x1,y1) = (3,-1)\r\n" );
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document.write( "y + 1 = (-1/2)(x - 3)\r\n" );
document.write( "y + 1 = (-1/2)x + 3/2\r\n" );
document.write( " y = (-1/2)x + 1/2\r\n" );
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document.write( "Now we will draw that line (in purple), and we have\r\n" );
document.write( "
\r\n" );
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document.write( "The brown and purple lines intersect at the center,\r\n" );
document.write( "so we find their point of intersection by solving\r\n" );
document.write( "the two equations as a system:\r\n" );
document.write( "\r\n" );
document.write( " y = -2x + 2\r\n" );
document.write( " y = (-1/2)x + 1/2\r\n" );
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document.write( "To make things easier we multiply the second\r\n" );
document.write( "equation through by 2\r\n" );
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document.write( " y = -2x + 2\r\n" );
document.write( " 2y = -x + 1\r\n" );
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document.write( "Can you solve this syetem by the substitution method?\r\n" );
document.write( "If not post again asking how to.\r\n" );
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document.write( "The solution is x = 1, y = 0\r\n" );
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document.write( "So the center of the circle is (1,0)\r\n" );
document.write( "\r\n" );
document.write( "Edwin
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document.write( " \n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
