document.write( "Question 698810: write the equation of the ellipse 4x^2+9y^2-64x+162y+949=0 in standard form. \n" ); document.write( "
Algebra.Com's Answer #431055 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "4x² + 9y² - 64x + 162y + 949 = 0\r\n" );
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document.write( "We want to get it in one of these forms:\r\n" );
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document.write( " or  \r\n" );
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document.write( "4x² + 9y² - 64x + 162y + 949 = 0\r\n" );
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document.write( "Swap the second and third terms to get the y-terms \r\n" );
document.write( "together and the x terms together.  Also add -949 \r\n" );
document.write( "to both sides:\r\n" );
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document.write( "      4x² - 64x + 9y² + 162y = -949 \r\n" );
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document.write( "Factor just the 4 out of the first two terms on the \r\n" );
document.write( "left, and just the 9 out of the last two terms on\r\n" );
document.write( "the left:\r\n" );
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document.write( "   4(x² - 16x) + 9(y² + 18y) = -949\r\n" );
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document.write( "We find what numbers we must add inside\r\n" );
document.write( "those parentheses to complete the square.\r\n" );
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document.write( "Multiply the coefficient of x in the first parentheses,\r\n" );
document.write( "which is -16 by.  Get -8.  The square that and\r\n" );
document.write( "get (-8)² = +64.  That's what we add inside the first \r\n" );
document.write( "parentheses.\r\n" );
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document.write( "Multiply the coefficient of y in the second parentheses,\r\n" );
document.write( "which is 18 by.  Get 9.  The square that and\r\n" );
document.write( "get 9² = +81.  That's what we add inside the second \r\n" );
document.write( "parentheses.\r\n" );
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document.write( "   4(x² - 16x + 64) + 9(y² + 18y + 81) = -949 + 256 + 729\r\n" );
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document.write( "Notice that we added 256 on the right because when +64 \r\n" );
document.write( "is added inside the first parentheses, since there is a 4\r\n" );
document.write( "coefficient before the entire first parentheses, it really \r\n" );
document.write( "amounts to adding 4×64 or 256 to the left side, so we have \r\n" );
document.write( "to do the same to the right side.\r\n" );
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document.write( "Notice also that we added 729 on the right because when +81 \r\n" );
document.write( "is added inside the second parentheses, since there is a 9\r\n" );
document.write( "coefficient before the entire first parentheses, it really \r\n" );
document.write( "amounts to adding 9×81 or 729 to the left side, so we have \r\n" );
document.write( "to do the same to the right side.\r\n" );
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document.write( "Now we factor the two parentheses and combine the terms on\r\n" );
document.write( "the right side:\r\n" );
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document.write( "     4(x - 8)(x - 8) + 9(y + 9)(y + 9) = 36\r\n" );
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document.write( "                 4(x - 8)² + 9(y + 9)² = 36\r\n" );
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document.write( "Divide through by 36 to get 1 on the right:\r\n" );
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document.write( "                  +  = \r\n" );
document.write( "               \r\n" );
document.write( "                  +  = \r\n" );
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document.write( "That's the form  \r\n" );
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document.write( "where (h,k) = (8,-9), a = 6, b = 2 and so the graph is\r\n" );
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document.write( "        \r\n" );
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document.write( "Edwin
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