Algebra.Com's Answer #696344 by ikleyn(52803)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "1. The center of the ellipse is at (0,0).\r\n" );
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document.write( " The major axis is y-axis x= 0;\r\n" );
document.write( " Hence, the minor axis is x-axis y= 0.\r\n" );
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document.write( "2. The canonical equation of this ellipse is \r\n" );
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document.write( "  = 1 (it is written in the form to fit the fact that the major axis is y-axis: a > b > 0)\r\n" );
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document.write( " Since the point (3,2) is on the ellipse, it implies \r\n" );
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document.write( "  = 1, or\r\n" );
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document.write( "  =  . (1)\r\n" );
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document.write( "3. The focal distance is 2c = 2 - (-2) = 4.\r\n" );
document.write( " Hence, the linear eccentricity c =  = 2. It means that\r\n" );
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document.write( "  = 4. (2)\r\n" );
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document.write( "4. Thus you have two equations (1) and (2) to determine \"a\" and \"b\".\r\n" );
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document.write( " You can simplify writing and solving by introducing new variables x =  and y =  :\r\n" );
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document.write( " 9x + 4y = xy (3) instead of (1), and\r\n" );
document.write( " x - y = 4. (4) instead of (2)\r\n" );
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document.write( "The setup is done.\r\n" );
document.write( "Now it is simple arithmetic to solve it and to get \"a\" and \"b\" at the end.\r\n" );
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document.write( "4. From (4), x = 4 + y, Substitute it into (3). You will get\r\n" );
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document.write( " 9(4+y) + 4y = (4+y)*y,\r\n" );
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document.write( " 36 + 9y + 4y = 4y + y^2 ---> y^2 -9y - 36 = 0 ---->  =  =  .\r\n" );
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document.write( " Only positive root works: y = 12. So, = 12 and b =  .\r\n" );
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document.write( " Then a^2 = b^2 + 4 = 12 + 4 = 16 and a =  = 4.\r\n" );
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document.write( " Thus semi-axes are 4 (vertical) and (horizontal).\r\n" );
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document.write( " The equation for the ellipse is\r\n" );
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document.write( "  +  = 1.\r\n" );
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document.write( "As a prerequisite, see the lesson\r
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document.write( " - Ellipse definition, canonical equation, characteristic points and elements \r
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document.write( "in this site.\r
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document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r
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document.write( " ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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document.write( "The referred lesson is the part of this online textbook under the topic
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document.write( "\"Conic sections: Ellipses. Definition, major elements and properties. Solved problems\".\r
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