document.write( "Question 707202: Write the standrad ang general equation of a circle through (2,1) and (3,5) and having its center on the line 8x + 5y = 8.
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Algebra.Com's Answer #435603 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
Write the standard and general equation of a circle through (2,1) and (3,5) and having its center on the line 8x + 5y = 8.\r
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document.write( "There is more than one way to do this, but let's do it the\r\n" );
document.write( "way the tutor above said to do it:\r\n" );
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document.write( "The standard equation of a 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 line connecting (2,1) amd (3,5) is a chord of the circle. \r\n" );
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document.write( "So we can use the fact that the perpendicular bisector of a chord \r\n" );
document.write( "passes through the center of the circle. That's the red line below\r\n" );
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document.write( "We get the slope of the green chord using the slope formula:\r\n" );
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document.write( "m = \"%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"\r\n" );
document.write( "m = \"%285-1%29%2F%283-2%29\"\r\n" );
document.write( "m = \"4%2F1\"\r\n" );
document.write( "m = 4\r\n" );
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document.write( "So the slope of the red perpendicular bisector of the green\r\n" );
document.write( "chord is it's \"negative reciprocal\" or \"-1%2F4\".\r\n" );
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document.write( "We get the midpoint of the green chord using the midpoint\r\n" );
document.write( "formula:\r\n" );
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document.write( "Midpoint = \r\n" );
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document.write( "Midpoint = \"%28matrix%281%2C3%2C%282%2B3%29%2F2%2C+%22%2C%22%2C%281%2B5%29%2F2%29%29\"\r\n" );
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document.write( "Midpoint = \"%28matrix%281%2C3%2C5%2F2%2C+%22%2C%22%2C6%2F2%29%29\" \r\n" );
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document.write( "Midpoint = \"%28matrix%281%2C3%2C5%2F2%2C+%22%2C%22%2C3%29%29\"\r\n" );
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document.write( "So the red perpendicular bisector of the chord is the line with\r\n" );
document.write( "slope \"-1%2F4\" that goes through the point \"%28matrix%281%2C3%2C5%2F2%2C+%22%2C%22%2C3%29%29\"\r\n" );
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document.write( "We use the point slope formula:\r\n" );
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document.write( "y - y1 = m(x - x1)\r\n" );
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document.write( "y - 3 = \"-1%2F4\"(x - 5/2)\r\n" );
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document.write( "y - 3 = \"-1%2F4\"x + 5/8\r\n" );
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document.write( "Multiply through by 8\r\n" );
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document.write( "8y - 24 = -2x + 5\r\n" );
document.write( "2x + 8y = 29\r\n" );
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document.write( "So to find the center we solve the system\r\n" );
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document.write( "\"system%288x+%2B+5y+=+8%2C2x+%2B+8y+=+29%29\"\r\n" );
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document.write( "and get the point (x,y) = (\"-3%2F2\",4)\r\n" );
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document.write( "That's the center (h,k) = (\"-3%2F2\",4)\r\n" );
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document.write( "So we can draw in the circle:\r\n" );
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document.write( "So the equation of the circle, so far is\r\n" );
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document.write( "(x-(\"-3%2F2\")² + (y-4)² = r²\r\n" );
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document.write( "(x+\"3%2F2\")² + (y-4)² = r²\r\n" );
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document.write( "And since we know it goes through (2,1) we can\r\n" );
document.write( "substitute that point:\r\n" );
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document.write( "(2+\"3%2F2\")² + (1-4)² = r²\r\n" );
document.write( "(\"4%2F2\"+\"3%2F2\")² + (-3)² = r²\r\n" );
document.write( "(\"7%2F2\")² + 9 = r²\r\n" );
document.write( "\"49%2F4\" + 9 = r²\r\n" );
document.write( "\"49%2F4\" + \"36%2F4\" = r²\r\n" );
document.write( "\"85%2F4\" = r²\r\n" );
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document.write( "So the standard equation of the circle is\r\n" );
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document.write( "(x+\"3%2F2\")² + (y-4)² = \"85%2F4\"\r\n" );
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document.write( "To find the general equatrion of the circle,\r\n" );
document.write( "nultiply that out:\r\n" );
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document.write( "x² + 3x + \"9%2F4\" + y² - 8y + 16 = \"85%2F4\"\r\n" );
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document.write( "Multiply through by 4\r\n" );
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document.write( "4x² + 12x + 9 + 4y² - 32y + 64 = 85\r\n" );
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document.write( "4x² + 12x + 73 + 4y² - 32y = 85\r\n" );
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document.write( "4x² + 4y² + 12x - 32y - 12 = 0\r\n" );
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document.write( "Divide through by 4\r\n" );
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document.write( "x² + y² + 3x - 8y - 3 = 0\r\n" );
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document.write( "That's the general equation of the circle.\r\n" );
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document.write( "Edwin
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