document.write( "Question 832262: Find the General Equation of the circle\r
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Algebra.Com's Answer #502012 by math-vortex(648) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "Find the general equation for the circle that passes through the points (3,0) , (4,2) and (0,1).\r\n" );
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document.write( "A SOLUTION:\r\n" );
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document.write( "The general equation for a circle has the form,\r\n" );
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document.write( "x^2 + y^2 + Dx + Ey + F = 0\r\n" );
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document.write( "We do not know the values of D, E, or F, but we do know three points on the circle. These ordered pairs \r\n" );
document.write( "must satisfy our equation.\r\n" );
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document.write( "We will substitute each of the (x,y) pairs into the general equation formula. Then we will have a system of\r\n" );
document.write( "three equations in three variables (D, E, F). We will solve this system in order to find the equation for \r\n" );
document.write( "the circle.\r\n" );
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document.write( "For the ordered pair (3,0): Substitute 3 for x and 0 for y in the general equation\r\n" );
document.write( "x^2 + y^2 + Dx + Ey + F = 0\r\n" );
document.write( "(3)^2 + (0)^2 + D(3) + E(0) + F = 0\r\n" );
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document.write( "Simplify.\r\n" );
document.write( "9 + 3D + F = 0\r\n" );
document.write( "3D + F = -9\r\n" );
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document.write( "For the ordered pair (4,2):\r\n" );
document.write( "(4)^2 + (2)^2 + D(4) + E(2) + F = 0\r\n" );
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document.write( "Simplify.\r\n" );
document.write( "16 + 4 + 4D + 2E + F = 0\r\n" );
document.write( "4D + 2E + F = -20\r\n" );
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document.write( "For the ordered pair (0,1): \r\n" );
document.write( "(0)^2 + (1)^2 + D(0) + E(1) + F = 0\r\n" );
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document.write( "Simplify.\r\n" );
document.write( "1 + E + F = 0\r\n" );
document.write( "E + F = -1\r\n" );
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document.write( "Our system of three equations is\r\n" );
document.write( "3D + F = -9\r\n" );
document.write( "4D + 2E + F =-20\r\n" );
document.write( "E + F = -1\r\n" );
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document.write( "Subtract Equation 3 from Equation 1 to eliminate F.\r\n" );
document.write( " 3D + F = -9\r\n" );
document.write( "-[E + F = -1]\r\n" );
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document.write( "3D - E = -8\r\n" );
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document.write( "Solve this equation for E.\r\n" );
document.write( "E = 3D + 8\r\n" );
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document.write( "Solve the first equation for F.\r\n" );
document.write( "3D + F = -9\r\n" );
document.write( "F = -3D - 9\r\n" );
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document.write( "Substitute 3D+8 for E  and -3D-9 for F in the second equation.\r\n" );
document.write( "4D + 2E + F =-20\r\n" );
document.write( "4D + 2(3D + 8) + (-3D - 9) = -20\r\n" );
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document.write( "Solve for D.\r\n" );
document.write( "4D + 6D + 16 - 3D - 9 = -20\r\n" );
document.write( "7D + 7 = -20\r\n" );
document.write( "7D = -27\r\n" );
document.write( "D = -27/7\r\n" );
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document.write( "Substitute -27/7 for D in the first equation.\r\n" );
document.write( "3D + F = -9\r\n" );
document.write( "3(-27/7) + F = -9\r\n" );
document.write( "(-81/7) + F = -9\r\n" );
document.write( "F = -63/7 + 81/7 \r\n" );
document.write( "F = 18/7\r\n" );
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document.write( "Substitute 18/7 for F in the third equation\r\n" );
document.write( "E + F = -1\r\n" );
document.write( "E + (18/7) = -1\r\n" );
document.write( "E = -25/7\r\n" );
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document.write( "Substitute -27/7 for D, -25/7 for E, and 18/7 for F in the general equation.\r\n" );
document.write( "x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0\r\n" );
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document.write( "The equation in general form is\r\n" );
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document.write( "Check your equation with the three given points. Each pair must make this equation true.\r\n" );
document.write( "For (3,0):\r\n" );
document.write( "x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0\r\n" );
document.write( "(3)^2 + (0)^2 + (-27/7)(3) + (-25/7)(0) + (18/7) = 0\r\n" );
document.write( "9 - 81/7 + 18/7 = 0\r\n" );
document.write( "(63 - 81 +18)/7 = 0 \r\n" );
document.write( "0/7 = 0 \r\n" );
document.write( "0 = 0 Check!\r\n" );
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document.write( "For (4,2):\r\n" );
document.write( "x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0\r\n" );
document.write( "(4)^2 + (2)^2 + (-27/7)(4) + (-25/7)(2) + (18/7) = 0\r\n" );
document.write( "16 + 4 - 108/7 - 50/7 + 18/7 = 0\r\n" );
document.write( "(140 - 108 - 50 + 18)/7 = 0 \r\n" );
document.write( "0/7 = 0  \r\n" );
document.write( "0 = 0 Check!\r\n" );
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document.write( "For (0,1):\r\n" );
document.write( "x^2 + y^2 + (-27/7)x + (-25/7)y + (18/7) = 0\r\n" );
document.write( "(0)^2 + (1)^2 + (-27/7)(0) + (-25/7)(1) + (18/7) = 0\r\n" );
document.write( "1 - 25/7 + 18/7 = 0\r\n" );
document.write( "(7 - 25 + 18)/7 = 0 \r\n" );
document.write( "0/7 = 0 \r\n" );
document.write( "0 = 0 Check!\r\n" );
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document.write( "Hope this helps! Feel free to email if you have any questions about the solution.\r\n" );
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document.write( "Good luck with your math,\r\n" );
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document.write( "Mrs. F\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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