document.write( "Question 36377: What is the y-coordinate of the center of the circle that passes through (-1,2),
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Algebra.Com's Answer #22281 by AnlytcPhil(1806)\"\" \"About 
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What is the y-coordinate of the center of the circle that\r\n" );
document.write( "passes through (-1,2), (3,2), and (5,4)?\r\n" );
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document.write( "The general equation of a circle is\r\n" );
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document.write( "x² + y² + Dx + Ey + F = 0\r\n" );
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document.write( "Substituting first point, (-1,2), i.e.,  x = -1, y = 2\r\n" );
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document.write( "(-1)² + (2)² + D(-1) + E(2) + F = 0\r\n" );
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document.write( "            1 + 4 - D + 2E + F =  0\r\n" );
document.write( "                5 - D + 2E + F =  0\r\n" );
document.write( "                   -D + 2E + F = -5\r\n" );
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document.write( "Substituting second point, (3,2), i.e.,  x = 3, y = 2\r\n" );
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document.write( "(3)² + (2)² + D(3) + E(2) + F = 0\r\n" );
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document.write( "            9 + 4 + 3D + 2E + F =  0\r\n" );
document.write( "               13 + 3D + 2E + F =  0\r\n" );
document.write( "                    3D + 2E + F = -13\r\n" );
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document.write( "Substituting third point, (5,4), i.e.,  x = 5, y = 4\r\n" );
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document.write( "(5)² + (4)² + D(5) + E(4) + F = 0\r\n" );
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document.write( "           25 + 16 + 5D + 4E + F =   0\r\n" );
document.write( "                41 + 5D + 4E + F =   0\r\n" );
document.write( "                     5D + 4E + F = -41\r\n" );
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document.write( "That gives you a system of three equations in \r\n" );
document.write( "three unknowns:\r\n" );
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document.write( " -D + 2E + F =  -5\r\n" );
document.write( " 3D + 2E + F = -13\r\n" );
document.write( " 5D + 4E + F = -41\r\n" );
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document.write( "D = -2, E = -12, F = 17\r\n" );
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document.write( "The general form the equation of the circle is\r\n" );
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document.write( "                    x² + y² - 2x - 12y + 17 = 0\r\n" );
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document.write( "Now we must get it in standard form\r\n" );
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document.write( "(x - h)² + (y - k)² = r²\r\n" );
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document.write( "                    x² + y² - 2x - 12y + 17 = 0\r\n" );
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document.write( "                         x² - 2x + y² - 12y = -17\r\n" );
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document.write( "                     (x² - 2x) + (y² - 12y) = -17\r\n" );
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document.write( "Complete squares:\r\n" );
document.write( "Multiply coeff's of x and y by 1/2, square the results,\r\n" );
document.write( "add inside parentheses and also to right side.\r\n" );
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document.write( "            (x² - 2x + 1) + (y² - 12y + 36) = -17 + 1 + 36\r\n" );
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document.write( "Factor the trinomials in parentheses, combine numbers \r\n" );
document.write( "on right:\r\n" );
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document.write( "                        (x - 1)² + (y - 6)² = 20\r\n" );
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document.write( "Compare to              (x - h)² + (y - k)² = r²\r\n" );
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document.write( "h = 1, k = 6, r² = 20, so center is (1, 6),\r\n" );
document.write( "              __     _ \r\n" );
document.write( "radius = r = Ö20 = 2Ö5 \r\n" );
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document.write( "You were asked only for the y-coordinate of the center,\r\n" );
document.write( "which is k = 6\r\n" );
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document.write( "Edwin McCravy\r\n" );
document.write( "AnlytcPhil@aol.com

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