document.write( "Question 1024265: find the center,foci and semi-axes of the ellipse x^2+4y^2=4x+8y \n" ); document.write( "
Algebra.Com's Answer #639720 by Edwin McCravy(20060) $\"\"$ $\"About$
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find the center,foci and semi-axes of the ellipse
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document.write( "You need to know the two standard forms of ellipses:\r\n" );
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document.write( " which is the equation\r\n" );
document.write( "of an ellipse that looks like this: \r\n" );
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document.write( "and\r\n" );
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document.write( " which is the equation\r\n" );
document.write( "of an ellipse that looks like this: \r\n" );
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document.write( "We can tell them apart because \"a\" is always a larger \r\n" );
document.write( "number than \"b\".\r\n" );
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document.write( "\"a\" is the length of the semi-major axis, and \"b\" is the\r\n" );
document.write( "length of the semi-minor axis.  The center is (h,k).  The\r\n" );
document.write( "vertices are the end points of the major axis. They are \r\n" );
document.write( "\"a\" units on each side of the center.  The co-vertices are \r\n" );
document.write( "the end points of the minor axis. They are respectively \"a\" \r\n" );
document.write( "and \"b\" units on each side of the center.\r\n" );
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document.write( "The foci are points on the major axis which are \"c\" units\r\n" );
document.write( "on each side of the center, where \"c\" is calculated by the\r\n" );
document.write( "formula:  \r\n" );
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document.write( "We must get the equation of an ellipse into one of the above\r\n" );
document.write( "standard forms:\r\n" );
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document.write( "To the side, multiply the coefficient of x,\r\n" );
document.write( "which is -4, by 1/2, which gives -2.  Then\r\n" );
document.write( "square -2 getting +4.  Add that inside the\r\n" );
document.write( "first parentheses and to the right side.\r\n" );
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document.write( "Also to the side, multiply the coefficient of y,\r\n" );
document.write( "which is -2, by 1/2, which gives -1.  Then\r\n" );
document.write( "square -1 getting +1.  Add that inside the\r\n" );
document.write( "first parentheses and to the right side. However\r\n" );
document.write( "since there is a +4 preceding that parentheses,\r\n" );
document.write( "adding +1 inside that parentheses is the same as\r\n" );
document.write( "adding +4 to the left side, so we add +4 to the right\r\n" );
document.write( "side:\r\n" );
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document.write( "We factor each parentheses and combine terms on the\r\n" );
document.write( "right:\r\n" );
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document.write( "Since the factorization results in the products of\r\n" );
document.write( "a binomial times itself, we can write each as the\r\n" );
document.write( "square of a binomial:\r\n" );
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document.write( "Next we get 1 on the right side by dividing through by 8\r\n" );
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document.write( "We divide top and bottom of the second fraction \r\n" );
document.write( "by 4 getting .  So we have:\r\n" );
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document.write( "Since the denominator 8 under the first term is greater \r\n" );
document.write( "than the denominator 2 under the second term, we know\r\n" );
document.write( "that the ellipse looks like this: \r\n" );
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document.write( "We also know that the center is (2,1).  We know that \r\n" );
document.write( " and , and  and .\r\n" );
document.write( "Since we know that  is approximately 1.4, we have\r\n" );
document.write( "this graph with the center and the two vertices:\r\n" );
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document.write( "We only need to find the two foci.  We calculate \"c\":\r\n" );
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document.write( "The foci are the two points \r\n" );
document.write( " and ,\r\n" );
document.write( "the red points below:\r\n" );
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document.write( "Edwin
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