document.write( "Question 1089712: Ellipse\r
\n" ); document.write( "\n" ); document.write( "Hello, would you please help me?\r
\n" ); document.write( "\n" ); document.write( "Express 4y^2 + 9x^2 - 24y - 72x + 144 = 0
\n" ); document.write( "to standard equation, then graph the ellipse.
\n" ); document.write( "Thank you very much. It would be a big help. \n" ); document.write( "
Algebra.Com's Answer #704073 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Express 4y^2 + 9x^2 - 24y - 72x + 144 = 0
\n" ); document.write( "to standard equation, then graph the ellipse.
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document.write( "The method is completing the squares separately for x-terms and y-terms.\r\n" );
document.write( "I will do it step by step - trace my steps very attentively.\r\n" );
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document.write( "4y^2 + 9x^2 - 24y - 72x + 144 = 0\r\n" );
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document.write( "4y^2 + 9x^2 - 24y - 72x = -144\r\n" );
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document.write( "(9x^2 - 72x) + (4y^2 - 24y) = -144\r\n" );
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document.write( "(9x^2 - 2*3*12x + 12^2) + (4y^2 -2*2*6y + 6^2) = -144 + 12^2 + 6^2\r\n" );
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document.write( "(3x-12)^2 + (2y-6)^2 = -144 + 144 + 36 = 36   (*)\r\n" );
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document.write( "Now divide both sides of the equation (*) by 36. You will get\r\n" );
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document.write( "\"%283x-12%29%5E2%2F36\" + \"%282y-6%29%5E2%2F36\" = 1.\r\n" );
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document.write( "Make couple of steps further\r\n" );
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document.write( "\"%289%2A%28x+-+4%29%5E2%29%2F36\" + \"%284%2A%28y-3%29%5E2%29%2F36\" = 1\r\n" );
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document.write( "\"%28%28x+-+4%29%5E2%29%2F4\" + \"%28%28y-3%29%5E2%29%2F9\" = 1\r\n" );
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document.write( "\"%28%28x+-+4%29%5E2%29%2F2%5E2\" + \"%28%28y-3%29%5E2%29%2F3%5E2\" = 1    (**)\r\n" );
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document.write( "Equation (**) is the standard form of an ellipse equation.\r\n" );
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document.write( "The plot is shown below:\r\n" );
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document.write( "            Ellipse \"%28x-4%29%5E2%2F2%5E2\" + \"%28y-3%29%5E2%2F3%5E2\" = 1\r\n" );
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document.write( "The center of the ellipse is at the point (4,3).\r\n" );
document.write( "The major axis is vertical.  The ellipse is taller than wide.\r\n" );
document.write( "The major semi-axis is 3 units long.\r\n" );
document.write( "The minor semi-axis is 2 units long.\r\n" );
document.write( "The linear eccentricity is \"sqrt%283%5E2-2%5E2%29\" = \"sqrt%285%29\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "There are lessons in this site that explain everything about ellipses, their equations and elements,
\n" ); document.write( "about finding and identifying the ellipse elements via equations and vice versa,
\n" ); document.write( "as well as transforming general equations of ellipses to their standard forms.\r
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\n" ); document.write( "\n" ); document.write( "These lessons are\r
\n" ); document.write( "\n" ); document.write( "    - Ellipse definition, canonical equation, characteristic points and elements \r
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\n" ); document.write( "\n" ); document.write( "    - Standard equation of an ellipse\r
\n" ); document.write( "\n" ); document.write( "    - Identify elements of an ellipse given by its standard equation\r
\n" ); document.write( "\n" ); document.write( "    - Find the standard equation of an ellipse given by its elements\r
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\n" ); document.write( "\n" ); document.write( "    - General equation of an ellipse\r
\n" ); document.write( "\n" ); document.write( "    - Transform a general equation of an ellipse to the standard form by completing the square\r
\n" ); document.write( "\n" ); document.write( "    - Identify elements of an ellipse given by its general equation\r
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\n" ); document.write( "\n" ); document.write( "and you will not find the better place in the Internet to learn this subject.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic
\n" ); document.write( "\"Conic sections: Ellipses. Definition, major elements and properties. Solved problems\".\r
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