document.write( "Question 1101234: Given the equation 25x^2+y^2-100x+8y+91=0
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Algebra.Com's Answer #715957 by ikleyn(52776)\"\" \"About 
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\n" ); document.write( "Given the equation 25x^2+y^2-100x+8y+91=0
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document.write( "Technically speaking, you need to complete squares separately for x- and y- terms.\r\n" );
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document.write( "25x^2 + y^2 - 100x + 8y + 91 = 0     is equivalent to\r\n" );
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document.write( "(25x^2 - 100x) + (y^2 + 8y) = - 91   is equivalent to\r\n" );
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document.write( "25*(x^2 - 4x)  + (y^2 + 8y) = -91    is equivalent to\r\n" );
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document.write( "25*(x^2 - 4x + 4) + (y^2 + 8y + 16) = -91 + 100 + 16    is equivalent to\r\n" );
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document.write( "25*(x-2)^2 + (y+4)^2 = 25      (<<<---=== divide by 25 both sides ===--->>>)  is equivalent to\r\n" );
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document.write( "\"%28x-2%29%5E2%2F1%5E2\" + \"%28y%2B4%29%5E2%2F5%5E2\" = 1.\r\n" );
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document.write( "The last equation is the standard form equation of the ellipse \r\n" );
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document.write( "   - centered at the point (2,-4)\r\n" );
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document.write( "   - with the major axis parallel to y-axis\r\n" );
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document.write( "   - taller than wide\r\n" );
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document.write( "   - with the major semi-axis of the length of 5 in vertical   direction and\r\n" );
document.write( "          the minor semi-axis of the length of 1 in horizontal direction.\r\n" );
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document.write( "The linear eccentricity of the ellipse is  \"sqrt%285%5E2-1%5E2%29\" = \"sqrt%2824%29\" = \"2%2Asqrt%286%29\".\r\n" );
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document.write( "The    vertices are  (2,-4+5) = (2,1)   and  (2,-4-5) = (2,-9).\r\n" );
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document.write( "The co-vertices are  (2+1,-4) = (3,-4)  and  (2-1,-4) = (1,-4).\r\n" );
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document.write( "The foci are  (2,-4+\"2%2Asqrt%286%29\")  and  (2,-4-\"2%2Asqrt%286%29\").\r\n" );
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document.write( "The plot is THIS:\r\n" );
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document.write( " Ellipse \"%28x-2%29%5E2%2F1%5E2\" + \"%28y%2B4%29%5E2%2F5%5E2\" = 1\r\n" );
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\n" ); document.write( "\n" ); document.write( "See the lessons\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( "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 lesson is 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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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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