Algebra.Com's Answer #817270 by ikleyn(52799)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "Hello there,
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document.write( "I have been stuck on a math problem for hours and have no idea of how to find the solution.
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document.write( "Please help me solve the following equation: Find an equation in standard form for the ellipse
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document.write( "with center at the origin and passes through the points (5, 6) and (-3, 7.06).
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document.write( "Ellipse with the center at origin has the standard form equation\r\n" );
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document.write( "  +  = 1. (1)\r\n" );
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document.write( "The coefficients \"a\" and \"b\" of this equation are the ellipse's semi-axis lengths.\r\n" );
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document.write( "These coefficients are UNKNOWN now, and our task is to find them.\r\n" );
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document.write( "For it, we will use this info about the points (5,6) and (-3,7.06), that they belong to the ellipse.\r\n" );
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document.write( "We substitute coordinates of these points into equation (1): it will give us\r\n" );
document.write( "two equations, from which we will determine \"a\" and \"b\".\r\n" );
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document.write( "So, first point gives us THIS equation\r\n" );
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document.write( "  +  = 1. (2)\r\n" );
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document.write( "Second points gives us THIS equation\r\n" );
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document.write( "  +  = 1. (3)\r\n" );
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document.write( "From the first glance, these equations are non-linear relative \"a\" and \"b\".\r\n" );
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document.write( "But they are LINEAR relative  and  .\r\n" );
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document.write( "So, we consider these two equations as the system of linear equations for and .\r\n" );
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document.write( "We solve this system of equations. It can be done by different methods.\r\n" );
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document.write( "It is assumed that the student knows all appropriate methods of linear algebra: substitution, elimination, the determinant method.\r\n" );
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document.write( "I used the determinant methos - same as the Cramer's rule.\r\n" );
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document.write( "So, without submerging into details, I got these solutions: = 0.015013285 and = 0.017351885.\r\n" );
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document.write( "It gives me values for \"a\" and \"b\" : a =  = 8.16135248 and b =  = 7.591483715.\r\n" );
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document.write( "Finally, the standard form equations of the ellipse are\r\n" );
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document.write( "  +  = 1. (4)\r\n" );
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document.write( "This equation is the ANSWER to the problem's question.\r\n" );
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document.write( "Solved.\r
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document.write( "The last step, which should be done, is to check if the answer is correct.\r\n" );
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document.write( "For it, the coordinates of given points should be substituted and left side should be calculated.\r\n" );
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document.write( "I did this check for both points and CONFIRMED the answer.\r\n" );
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document.write( "Happy learning (!)\r
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document.write( "Which grade, which school, which college, which university, which country and which thinking center are you from ?\r
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