document.write( "Question 616919: find the center of the circle using these three points (-4,4),(-7,3),(-8,2) \n" ); document.write( "
Algebra.Com's Answer #388005 by EdwinParker(16) $\"\"$ $\"About$
You can put this solution on YOUR website!
(-4,4),(-7,3),(-8,2)
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document.write( "Substitute those three points for (x,y) into the standard \r\n" );
document.write( "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( "So we have this system of three equations in three unknowns:\r\n" );
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document.write( "(-4-h)² + (4-k)² = r²\r\n" );
document.write( "(-7-h)² + (3-k)² = r²\r\n" );
document.write( "(-8-h)² + (2-k)² = r²\r\n" );
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document.write( "Set the left sides of the first two equations equal:\r\n" );
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document.write( "(-4-h)² + (4-k)² = (-7-h)² + (3-k)² \r\n" );
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document.write( "Rearrange so as to have difference of squares on each side:\r\n" );
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document.write( "(-4-h)² - (-7-h)² = (3-k)² - (4-k)²\r\n" );
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document.write( "Factor both sides as the difference of squares:\r\n" );
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document.write( "[(-4-h) - (-7-h)][(-4-h) + (-7-h)] = [(3-k) - (4-k)][(3-k) + (4-k)]\r\n" );
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document.write( "Remove the inner parentheses:\r\n" );
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document.write( "[-4 - h + 7 + h][-4 - h - 7 - h] = [3 - k - 4 + k][3 - k + 4 - k]\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "[3][-11 - 2h] = [-1][7 - 2k]\r\n" );
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document.write( "-33 -6h = -7 + 2k\r\n" );
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document.write( "-6h - 2k = 26\r\n" );
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document.write( "Set the left sides of the first and third equations equal:\r\n" );
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document.write( "(-4-h)² + (4-k)² = (-8-h)² + (2-k)² \r\n" );
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document.write( "Rearrange so as to have difference of squares on each side:\r\n" );
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document.write( "(-4-h)² - (-8-h)² = (2-k)² - (4-k)²\r\n" );
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document.write( "Factor both sides as the difference of squares:\r\n" );
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document.write( "[(-4-h) - (-8-h)][(-4-h) + (-8-h)] = [(2-k) - (4-k)][(2-k) + (4-k)]\r\n" );
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document.write( "Remove the inner parentheses:\r\n" );
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document.write( "[-4 - h + 8 + h][-4 - h - 8 - h] = [2 - k - 4 + k][2 - k + 4 - k]\r\n" );
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document.write( "Simplify:\r\n" );
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document.write( "[4][-12 - 2h] = [-2][6 - 2k]\r\n" );
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document.write( "-48 - 8h = -12 + 4k\r\n" );
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document.write( "-8h - 4k = 36\r\n" );
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document.write( "Now we have the system of equations:\r\n" );
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document.write( "-6h - 2k = 26\r\n" );
document.write( "-8h - 4k = 36\r\n" );
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document.write( "Solve that system and get (h,k) = (-4,-1)\r\n" );
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document.write( "That's the center.  That's all you asked for.\r\n" );
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document.write( "But if you had been asked for the equation, or if you\r\n" );
document.write( "wanted to check your problem, you would also need to \r\n" );
document.write( "find the radius. \r\n" );
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document.write( "Substitute (h,k) = (-4,-1) in any one of the original \r\n" );
document.write( "3 equations.  I'll pick the first one:\r\n" );
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document.write( "      (-4-h)² + (4-k)² = r²\r\n" );
document.write( "(-4-(-4))² + (4-(-1))² = r²\r\n" );
document.write( "      (-4+4)² + (4+1)² = r²\r\n" );
document.write( "               0² + 5² = r²\r\n" );
document.write( "                    5² = r²\r\n" );
document.write( "                     5 = r\r\n" );
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document.write( "So the radius is 5, and so the equation of the circle is:\r\n" );
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document.write( "(x+4)² + (y+1)² = 5²\r\n" );
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document.write( "(x+4)² + (y+1)² = 25\r\n" );
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document.write( "Edwin
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