document.write( "Question 860401: Rotate the axes and eliminate the xy term of x^2 + 8xy + y^2 - 15 = 0. please include necessary detailed explanations if possible. thank you \n" ); document.write( "
Algebra.Com's Answer #518409 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Although most textbooks use x' and y', for convenience I\r\n" );
document.write( "will use capital X instead of x' and capital Y instead of y'.\r\n" );
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document.write( "The angle of rotation \"theta\" of the graph \r\n" );
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document.write( "Ax² + Bxy + Cy² + Dx + Ey + F = 0\r\n" );
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document.write( "necessary to transform it into an equation in X and Y which contains\r\n" );
document.write( "no term in XY\r\n" );
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document.write( "is calculated by \"tan%282theta%29=B%2F%28A-C%29\" or \"pi%2F4\" if A = C\r\n" );
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document.write( "Your equation is\r\n" );
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document.write( "x² + 8xy + y² - 15 = 0.\r\n" );
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document.write( "In this case A-1, B=8, C=1, D=0, E=0, F = -15\r\n" );
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document.write( "Since A=C=1 the required angle of rotation is \"theta=pi%2F4\"\r\n" );
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document.write( "The substitutions to make are \r\n" );
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document.write( "\"x=X%2Acos%28theta%29-Y%2Asin%28theta%29\"\r\n" );
document.write( "\"y=X%2Asin%28theta%29%2BY%2Acos%28theta%29\"\r\n" );
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document.write( "Since \"cos%28pi%2F4%29=sin%28pi%2F4%29=sqrt%282%29%2F2%29\", the substitutions are\r\n" );
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document.write( "\"x=X%28sqrt%282%29%2F2%29-Y%28sqrt%282%29%2F2%29\"\r\n" );
document.write( "\"y=X%28sqrt%282%29%2F2%29%2BY%28sqrt%282%29%2F2%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "\"x=expr%28sqrt%282%29%2F2%29%28X-Y%29\"\r\n" );
document.write( "\"y=expr%28sqrt%282%29%2F2%29%28X%2BY%29\"\r\n" );
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document.write( "or\r\n" );
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document.write( "x² + 8xy + y² - 15 = 0 becomes\r\n" );
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document.write( "Multiply through by 2\r\n" );
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document.write( "\"%28X%5E2-2XY%2BY%5E2%29%2B8%28X%5E2-Y%5E2%29%2B%28X%5E2%2B2XY%2BY%5E2%29-30=0\"\r\n" );
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document.write( "\"X%5E2-2XY%2BY%5E2%2B8X%5E2-8Y%5E2%2BX%5E2%2B2XY%2BY%5E2-30=0\"\r\n" );
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document.write( "\"10X%5E2-6Y%5E2-30=0\"\r\n" );
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document.write( "\"10X%5E2-6Y%5E2=30\"\r\n" );
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document.write( "Divide through by 30 to get 1 on the right side:\r\n" );
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document.write( "\"X%5E2%2F3-Y%5E2%2F5=1\"\r\n" );
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document.write( "Edwin
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