document.write( "Question 748607: the circle passes through the point (0,0), (5,0) and (3,3) \n" ); document.write( "
Algebra.Com's Answer #455566 by AnlytcPhil(1810)\"\" \"About 
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the circle passes through the point (0,0), (5,0) and (3,3)
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document.write( "You can do this either by using the general form or the standard form.\r\n" );
document.write( "I'll first do it with the general form:\r\n" );
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document.write( "x² + y² + Dx + Ey + F = 0\r\n" );
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document.write( "Substitute (x,y) = (0,0)\r\n" );
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document.write( "0² + 0² + D(0) + E(0) + F = 0\r\n" );
document.write( "                        F = 0\r\n" );
document.write( "So we have\r\n" );
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document.write( "x² + y² + Dx + Ey + 0 = 0\r\n" );
document.write( "    x² + y² + Dx + Ey = 0\r\n" );
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document.write( "Substitute (x,y) = (5,0)\r\n" );
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document.write( "5² + 0² + D(5) + E(0) = 0\r\n" );
document.write( "              25 + 5D = 0\r\n" );
document.write( "              25 + 5D = 0\r\n" );
document.write( "                   5D = -25\r\n" );
document.write( "                    D = -5\r\n" );
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document.write( "So we have:\r\n" );
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document.write( "    x² + y² + Dx + Ey = 0\r\n" );
document.write( "    x² + y² - 5x + Ey = 0\r\n" );
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document.write( "Substitute (x,y) = (3,3)\r\n" );
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document.write( "        x² + y² - 5x + Ey = 0\r\n" );
document.write( "(3)² + (3)² - 5(3) + E(3) = 0\r\n" );
document.write( "         9 + 9 - 15 + 3E = 0\r\n" );
document.write( "                  3 + 3E = 0\r\n" );
document.write( "                      3E = -3\r\n" );
document.write( "                       E = -1  \r\n" );
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document.write( "        x² + y² - 5x + Ey = 0\r\n" );
document.write( "         x² + y² - 5x - y = 0\r\n" );
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document.write( "You can also use the standard form equation for a circle:\r\n" );
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document.write( "      (x - h)² + (y - k)² = r²\r\n" );
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document.write( "Substitute (x,y) = (0,0)\r\n" );
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document.write( "      (0 - h)² + (0 - k)² = r²\r\n" );
document.write( "                  h² + k² = r²\r\n" );
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document.write( "Substitute (x,y) = (5,0)\r\n" );
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document.write( "      (5 - h)² + (0 - k)² = r²\r\n" );
document.write( "       25 - 10h + h² + k² = r²\r\n" );
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document.write( "Substitute (x,y) = (3,3)\r\n" );
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document.write( "      (3 - h)² + (3 - k)² = r²\r\n" );
document.write( "9 - 6h + h² + 9 - 6k + k² = r²\r\n" );
document.write( "   18 - 6h - 6k + h² + k² = r²\r\n" );
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document.write( "So we have this system of equations:\r\n" );
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document.write( "                  h² + k² = r²\r\n" );
document.write( "       25 - 10h + h² + k² = r²\r\n" );
document.write( "   18 - 6h - 6k + h² + k² = r² \r\n" );
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document.write( "If we subtract the first equation term by term from each of the\r\n" );
document.write( "other two equations we have:\r\n" );
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document.write( "       25 - 10h = 0\r\n" );
document.write( "   18 - 6h - 6k = 0\r\n" );
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document.write( "Solving the first for h\r\n" );
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document.write( "       25 - 10h = 0\r\n" );
document.write( "           -10h = -25 \r\n" );
document.write( "              h = \"%28-25%29%2F%28-10%29\"\r\n" );
document.write( "              h = \"5%2F2\"\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "   18 - 6h - 6k = 0\r\n" );
document.write( "   18 - 6\"%285%2F2%29\" - 6k = 0\r\n" );
document.write( "   18 - 15 - 6k = 0\r\n" );
document.write( "         3 - 6k = 0\r\n" );
document.write( "            -6k = -3\r\n" );
document.write( "              k = \"%28-3%29%2F%28-6%29\"\r\n" );
document.write( "              k = \"1%2F2\"\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "        h² + k² = r²\r\n" );
document.write( "        \"%285%2F2%29%5E2\"\"%22%22%2B%22%22\"\"%281%2F2%29%5E2\"\"%22%22=%22%22\"\"r%5E2\"\r\n" );
document.write( "        \"25%2F4\"\"%22%22%2B%22%22\"\"1%2F4\"\"%22%22=%22%22\"\"r%5E2\"\r\n" );
document.write( "                             \"26%2F4\"\"%22%22=%22%22\"\"r%5E2\"\r\n" );
document.write( "                             \"13%2F2\"\"%22%22=%22%22\"\"r%5E2\"\r\n" );
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document.write( "So the equation\r\n" );
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document.write( "      (x - h)² + (y - k)² = r² \r\n" );
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document.write( "becomes\r\n" );
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document.write( "      (x - \"5%2F2\")² + (y - \"1%2F2\")² = \"13%2F2\"\r\n" );
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document.write( "It was harder but mainly because the center and radius were fractions.\r\n" );
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document.write( "Here is the graph.  Notice that the circle passes through the\r\n" );
document.write( "given points and has center (\"5%2F2\",\"1%2F2\") or (2.5,.5)\r\n" );
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document.write( "The radius is \"sqrt%2813%2F2%29\" ≈ 2.55 \r\n" );
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document.write( "Edwin
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