\r\n" );
document.write( "x²-y²-2x+4y-3=0\r\n" );
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document.write( "We complete the square. (This is a degenerate hyperbola\r\n" );
document.write( "which is degenerated into its two asymptotes. Its center,\r\n" );
document.write( "foci and vertices all coincide at the intersection of the\r\n" );
document.write( "two asymptotes)\r\n" );
document.write( "\r\n" );
document.write( "x²-2x-y²+4y=3\r\n" );
document.write( "\r\n" );
document.write( "(x²-2x)-(y²-4y)=3\r\n" );
document.write( "\r\n" );
document.write( "(x²-2x+1)-(y²-4y+4)=3+1-4\r\n" );
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document.write( "(x-1)²-(y-2)² = 0\r\n" );
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document.write( "Factor as the difference of two squares:\r\n" );
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document.write( "[(x-1)-(y-2)][(x-1)+(y-2)] = 0\r\n" );
document.write( "\r\n" );
document.write( "[x-1-y+2][x-1+y-2] = 0\r\n" );
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document.write( "(x-y+1)(x+y-3) = 0\r\n" );
document.write( "\r\n" );
document.write( "x-y+1 = 0; x+y-3 = 0\r\n" );
document.write( " -y = -x-1; y = -x+3\r\n" );
document.write( " y = x+1; \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Those are the equations of the two lines.\r\n" );
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document.write( "We graph them and plot the point which the circle goes through\r\n" );
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document.write( "
\r\n" );
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document.write( "Since two diameters of the same circle must intersect at the center\r\n" );
document.write( "of the circle. The center of the circle must be the intersection of\r\n" );
document.write( "those two lines. So we sketch the circle below: \r\n" );
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document.write( "
\r\n" );
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document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We solve them as a system to find their point\r\n" );
document.write( "of intersection:\r\n" );
document.write( "\r\n" );
document.write( "y = -x+3\r\n" );
document.write( "y = x+1\r\n" );
document.write( "\r\n" );
document.write( "-x+3 = x+1\r\n" );
document.write( " -2x = -2\r\n" );
document.write( " x = 1\r\n" );
document.write( "Substitute\r\n" );
document.write( "y = 1+1\r\n" );
document.write( "y=2\r\n" );
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document.write( "So they intersect at (1,2), so that's the center, and the radius is 1\r\n" );
document.write( "because the distance from the center to the point on the circle (1,1)\r\n" );
document.write( "is 1 unit. So the equation is\r\n" );
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document.write( "(x-h)²+(y-k)² = r² \r\n" );
document.write( "\r\n" );
document.write( "(x-1)²+(y-2)² = 1\r\n" );
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document.write( "If you like, you can multiply that out and get:\r\n" );
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document.write( "x²-2x+1+y²-2y+4 = 1\r\n" );
document.write( "\r\n" );
document.write( "or\r\n" );
document.write( "\r\n" );
document.write( " x²+y²-2x-2y+3 = 0 \r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
