document.write( "Question 767517: Identify the points of the ellipse 4x^2 + 5y^2 - 8x + 20y = -24. \n" ); document.write( "
Algebra.Com's Answer #467659 by Edwin McCravy(20060)\"\" \"About 
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document.write( "This conic appears at first to be an ellipse since there is\r\n" );
document.write( "no xy term, an x² and a y² term with different coefficients\r\n" );
document.write( "with the same sign when on the same side of the equation.  \r\n" );
document.write( "So we try to get it in standard form.  Either\r\n" );
document.write( "in the form \"%28x-h%29%5E2%2Fa%5E2%2B%28y-k%29%5E2%2Fb%5E2=1\" or \"%28x-h%29%5E2%2Fb%5E2%2B%28y-k%29%5E2%2Fa%5E2=1\" \r\n" );
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document.write( "   4x² + 5y² - 8x + 20y = -24\r\n" );
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document.write( "   4x² - 8x + 5y² + 20y = -24\r\n" );
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document.write( "4(x² - 2x) + 5(y² + 4y) = -24\r\n" );
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document.write( "To complete the square in the first parentheses,\r\n" );
document.write( "Multiply the coefficient of x, which is -2, by 1/2\r\n" );
document.write( "getting -1.  Then square -1, which gives +1. So we\r\n" );
document.write( "add +1 to the right of the first parentheses, and\r\n" );
document.write( "since there is a 4 multiplied by the first parentheses\r\n" );
document.write( "we add 4·1 or 4 to the right side of the equation:\r\n" );
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document.write( "4(x² - 2x + 1) + 5(y² + 4y) = -24 + 4\r\n" );
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document.write( "To complete the square in the second parentheses,\r\n" );
document.write( "Multiply the coefficient of y, which is 4, by 1/2\r\n" );
document.write( "getting 2.  Then square 2, which gives 4. So we\r\n" );
document.write( "add 4 to the right of the second parentheses, and\r\n" );
document.write( "since there is a 5 multiplied by the second parentheses\r\n" );
document.write( "we add 5·4 or 20 to the right side of the equation:\r\n" );
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document.write( "4(x² - 2x + 1) + 5(y² + 4y + 4) = -24 + 4 + 20\r\n" );
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document.write( "Factor the first parentheses as (x - 1)(x - 1) or (x - 1)²\r\n" );
document.write( "Factor the second parentheses as (x - 2)(x - 2) or (x - 2)²\r\n" );
document.write( "Combine the numbers on the right, getting 0.\r\n" );
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document.write( "4(x - 1)² + 5(y + 2) = 0\r\n" );
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document.write( "Oh oh.  This is an unusual situation because the right\r\n" );
document.write( "side came out to be 0.  So we cannot get 1 on the right\r\n" );
document.write( "side as we could if it were some other number.\r\n" );
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document.write( "This means that the graph is of an ellipse that is\r\n" );
document.write( "degenerated into a single point.  The graph of the\r\n" );
document.write( "equation is simply a single point (1,-2)\r\n" );
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document.write( "The graph is simply this single point (1,-2):\r\n" );
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document.write( "This equation is the equation of a single point.  A single point\r\n" );
document.write( "IS a conic section, because the vertex of a cone is a single point.\r\n" );
document.write( "A plane intersecting a cone at its vertex is a single point. \r\n" );
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document.write( "Edwin
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