document.write( "Question 454639: I need help with this problem:\r
\n" ); document.write( "\n" ); document.write( "Give the center and radius of the circle x^2+y^2+4x-2y=4 and then scatch its graph.\r
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Algebra.Com's Answer #312155 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "x² + y² + 4x - 2y = 4\r\n" );
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document.write( "Rearrange the terms and skip some spaces:\r\n" );
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document.write( "x² + 4x     + y² - 2y     = 4 \r\n" );
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document.write( "Take half the coefficient of x which is 4, get 2\r\n" );
document.write( "Square 2, get 4.  Add + 4 in the first space on \r\n" );
document.write( "the left and also add + 4 to the right side:\r\n" );
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document.write( "x² + 4x + 4 + y² - 2y     = 4 + 4\r\n" );
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document.write( "Take half the coefficient of y which is -2, get -1\r\n" );
document.write( "Square -1, get +1.  Add + 1 in the second space on\r\n" );
document.write( "the left and also add + 1 to the right side:\r\n" );
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document.write( "x² + 4x + 4 + y² - 2y + 1  = 4 + 4 + 1\r\n" );
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document.write( "Combine the terms on the right side\r\n" );
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document.write( "x² + 4x + 4 + y² - 2y + 1  = 9\r\n" );
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document.write( "Factor the first three terms on the left as\r\n" );
document.write( "(x + 2)(x + 2) or (x + 2)²\r\n" );
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document.write( "   (x + 2)² + y² - 2y + 1  = 9\r\n" );
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document.write( "Factor the last three terms on the left as\r\n" );
document.write( "(y - 1)(y - 1) or (y - 1)²\r\n" );
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document.write( "       (x + 2)² + (y - 1)² = 9\r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "      (x - h)² + (y - k)² = r²\r\n" );
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document.write( "-h = +2 so h = -2\r\n" );
document.write( "-k = -1 so k = 1\r\n" );
document.write( "r² = 9  so r = 3\r\n" );
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document.write( "So this is a circle with center (h,k) = (-2,1) \r\n" );
document.write( "and radius 3\r\n" );
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document.write( "So we get a compass and stick the sharp point \r\n" );
document.write( "on the green point below:\r\n" );
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document.write( "Then open out the compass to a radius of 3 and \r\n" );
document.write( "swing it around to draw this circle:\r\n" );
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document.write( "To check it we can see that the graph goes \r\n" );
document.write( "through these four points:\r\n" );
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document.write( "(-5,1), (-2,4), (1,1) and (-2,-2)\r\n" );
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document.write( "We check those points to see if they all \r\n" );
document.write( "satisfy the original equation:\r\n" );
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document.write( "        x² + y² + 4x - 2y = 4\r\n" );
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document.write( "Checking (-5,1)\r\n" );
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document.write( "5² + (-1)² + 4(-5) - 2(1) = 4\r\n" );
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document.write( "          25 + 1 - 20 - 2 = 4\r\n" );
document.write( "              26 - 20 - 2 = 4\r\n" );
document.write( "                    6 - 2 = 4\r\n" );
document.write( "                        4 = 4\r\n" );
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document.write( "That one does.\r\n" );
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document.write( "Checking (-2,4)\r\n" );
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document.write( "(-2)² + 4² + 4(-2) - 2(4) = 4\r\n" );
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document.write( "           4 + 16 - 8 - 8 = 4\r\n" );
document.write( "               20 - 8 - 8 = 4\r\n" );
document.write( "                   12 - 8 = 4\r\n" );
document.write( "                        4 = 4\r\n" );
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document.write( "That one does.\r\n" );
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document.write( "Checking (1,1)\r\n" );
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document.write( "    1² + 1² + 4(1) - 2(1) = 4\r\n" );
document.write( "            1 + 1 + 4 - 2 = 4\r\n" );
document.write( "                2 + 4 - 2 = 4\r\n" );
document.write( "                    6 - 2 = 4\r\n" );
document.write( "                        4 = 4\r\n" );
document.write( "That one does.\r\n" );
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document.write( "Checking (-2,-2)\r\n" );
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document.write( "(-2)² + (-2)² + 4(-2) - 2(-2) = 4\r\n" );
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document.write( "                4 + 4 - 8 + 4 = 4\r\n" );
document.write( "                    8 - 8 + 4 = 4\r\n" );
document.write( "                            4 = 4\r\n" );
document.write( "That one does too. So now we know we have the\r\n" );
document.write( "right circle.\r\n" );
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document.write( "Edwin
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