document.write( "Question 944741: Find the equation of the circle with center on the line X-5y+9=0, tangent to X-3y-11=0, and has radius √10 \n" ); document.write( "
Algebra.Com's Answer #576092 by Edwin McCravy(20065)\"\" \"About 
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document.write( "It appears that there are two solutions:\r\n" );
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document.write( "Let the center of the circle be (h,k).\r\n" );
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document.write( "Then the perpendicular distance from the center of the circle\r\n" );
document.write( "to the line to which it is tangent must equal to the radius of the circle.\r\n" );
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document.write( "That's the green lines.\r\n" );
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document.write( "The perpendicular distance from the point (x1,y1)\r\n" );
document.write( "to the line Ax+By+C=0 is\r\n" );
document.write( "d = \"abs%28Ax%5B1%5D%2BBy%5B1%5D%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29\"\r\n" );
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document.write( "Therefore, the perpendicular distance from ine point (h,k) to the line \r\n" );
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document.write( "      x-3y-11 = 0 \r\n" );
document.write( "is\r\n" );
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document.write( "\"abs%28h-3k-11%29%2Fsqrt%281%5E2%2B%28-3%29%5E2%29\"\r\n" );
document.write( "\"abs%28h-3k-11%29%2Fsqrt%281%2B9%29\"\r\n" );
document.write( "\"abs%28h-3k-11%29%2Fsqrt%2810%29\"\r\n" );
document.write( "It must equal to the radius \"sqrt%2810%29\" so\r\n" );
document.write( "\"abs%28h-3k-11%29%2Fsqrt%2810%29\"\"%22%22=%22%22\"\"sqrt%2810%29\"\r\n" );
document.write( "\"abs%28h-3k-11%29\"\"%22%22=%22%22\"\"10\"\r\n" );
document.write( "Since (h,k) is on the line x-5y+9=0\r\n" );
document.write( "x-5y+9=0\r\n" );
document.write( "h-5k+9=0\r\n" );
document.write( "h=5k-9\r\n" );
document.write( "Substitute in\r\n" );
document.write( "\"abs%28%285k-9%29-3k-11%29\"\"%22%22=%22%22\"\"10\"\r\n" );
document.write( "\"abs%285k-9-3k-11%29\"\"%22%22=%22%22\"\"10\"\r\n" );
document.write( "\"abs%282k-20%29\"\"%22%22=%22%22\"\"10\"\r\n" );
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document.write( "That breaks into two equations:\r\n" );
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document.write( "2k - 20 = 10     and    -(2k - 20) = 10\r\n" );
document.write( "     2k = 30     and      -2k + 20 = 10\r\n" );
document.write( "      k = 15     and           -2k = -10\r\n" );
document.write( "                                 k = 5\r\n" );
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document.write( "  h = 5k - 9                h = 5k - 9\r\n" );
document.write( "  h = 5(15) - 9             h = 5(5) - 9\r\n" );
document.write( "  h = 75 - 9                h = 25 - 9\r\n" );
document.write( "  h = 66                    h = 16\r\n" );
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document.write( "(h,k) = (66,15)             (h,k) = (16,5)\r\n" );
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document.write( "The equation of any circle is\r\n" );
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document.write( "(x-h)² + (y-k)² = r²\r\n" );
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document.write( "The radius r is given as \"sqrt%2810%29\", so r² = 10  \r\n" );
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document.write( "So the two circles' equations are:\r\n" );
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document.write( "(x-66)² + (y-16)² = 10   and  (x-16)² + (y-5)² = 10\r\n" );
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document.write( "Edwin
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