document.write( "Question 980038: Find an equation of the line containing the centers of the two circles:
\n" ); document.write( "x² + y² - 10x - 10y + 49 = 0 and x² + y² - 4x - 6y + 9 = 0.
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\n" ); document.write( "a. -2x - 3y + 5 = 0
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\n" ); document.write( "b. 2x + 3y + 5 = 0
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\n" ); document.write( "c. 8x - 7y + 5 = 0
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\n" ); document.write( "d. 2x - 3y + 5 = 0 \n" ); document.write( "

Algebra.Com's Answer #601250 by Edwin McCravy(20056) $\"\"$ $\"About$

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Find an equation of the line containing the centers of the two circles:
\n" ); document.write( "x² + y² - 10x - 10y + 49 = 0 and x² + y² - 4x - 6y + 9 = 0.
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\n" ); document.write( "a. -2x - 3y + 5 = 0
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\n" ); document.write( "b. 2x + 3y + 5 = 0
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\n" ); document.write( "c. 8x - 7y + 5 = 0
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\n" ); document.write( "d. 2x - 3y + 5 = 0\r
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document.write( "I'll show you two ways, an easy way and a harder way,\r\n" );
document.write( "though neither is hard:\r\n" );
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document.write( "Here's the easiest way:\r\n" );
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document.write( " x² + y² - 10x - 10y + 49 = 0\r\n" );
document.write( " x² + y² -  4x -  6y +  9 = 0\r\n" );
document.write( "-----------------------------\r\n" );
document.write( "           -6x -  4y + 40 = 0 \r\n" );
document.write( "or\r\n" );
document.write( "            3x +  2y - 20 = 0\r\n" );
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document.write( "Subtracting the two equations of a circle gives the \r\n" );
document.write( "equation of the radical axis of the two circles, which \r\n" );
document.write( "is perpendicular to the line through their centers.\r\n" );
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document.write( "The slope of any line whose equation is Ax+By+C=0 has \r\n" );
document.write( "slope -A/B, so the radical axis has slope -3/2 and a\r\n" );
document.write( "line perpendicular to it has slope 2/3.  As it turns\r\n" );
document.write( "out, only choice d has slope 2/3.  So the answer is d.\r\n" );
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document.write( "However I think your instructor would not accept this\r\n" );
document.write( "method although it is quite correct.  \r\n" );
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document.write( "Reason 1: There may have been more than one choice of \r\n" );
document.write( "equations of lines with slope 2/3, and you would not\r\n" );
document.write( "be able to determine which one it was.\r\n" );
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document.write( "Reason 2:  You may not have studied the radical axis \r\n" );
document.write( "of two circles. \r\n" );
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document.write( "----------------------------\r\n" );
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document.write( "The other way.\r\n" );
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document.write( "We find the centers of the two circles and find the equation \r\n" );
document.write( "of the line through them.\r\n" );
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document.write( "But there is an easy way to find the center of a circle\r\n" );
document.write( "whose equation is\r\n" );
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document.write( "x² + y² + Dx + Ey + F = 0\r\n" );
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document.write( "The center is simply (h,k) = \r\n" );
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document.write( "Therefore\r\n" );
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document.write( " x² + y² - 10x - 10y + 49 = 0 has center (5,5)\r\n" );
document.write( " x² + y² -  4x -  6y +  9 = 0 has center (2,3)\r\n" );
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document.write( "So the slope is \r\n" );
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document.write( "And equation:  y-3 = (x-2)\r\n" );
document.write( "              3y-9 = 2(x-2)\r\n" );
document.write( "              3y-9 = 2x-4\r\n" );
document.write( "          -2x+3y-5 = 0\r\n" );
document.write( "           2x-3y+5 = 0\r\n" );
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document.write( "Choice d. \r\n" );
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document.write( "Edwin

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