document.write( "Question 1149909: Common tangents are drawn to parabola y²=4x and ellipse 3x²+8y²=48 touching the parabola at A & B and the ellipse at C & D.Find the area of quadrilateral \n" ); document.write( "
Algebra.Com's Answer #771317 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We must find all the points of tangency. \r\n" );
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document.write( "I'll assume you haven't had calculus, it's a lots harder that way! lol. \r\n" );
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document.write( "Let the common tangent line have equation y=mx+b, solve the system\r\n" );
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document.write( "\"system%28y=mx%2Bb%2Cy%5E2=4x%29\" \r\n" );
document.write( "and find the discriminant of the resulting quadratic to be 1-bm. Set it equal to\r\n" );
document.write( "0, because there is only one solution at a tangent point.  Then solve the\r\n" );
document.write( "system:\r\n" );
document.write( "\"system%28y=mx%2Bb%2C3x%5E2%2B8y%5E2=48%29\" \r\n" );
document.write( "and find the discriminant of the resulting quadratic to be -b²+16m²+6.  Set it\r\n" );
document.write( "equal to 0.  Then solve the system\r\n" );
document.write( "\"system%281-bm=0%2C-b%5E2%2B16m%5E2%2B6=0%29\"\r\n" );
document.write( "You'll get\r\n" );
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document.write( "Substitute the positive values for b and m in y=mx+b, and solve simultaneously\r\n" );
document.write( "with the parabola and get: \r\n" );
document.write( "\"A=%28matrix%281%2C3%2C8%2C%22%2C%22%2C4sqrt%282%29%29%29\"  \r\n" );
document.write( "and you can tell by symmetry that \r\n" );
document.write( "\"B=%28matrix%281%2C3%2C8%2C%22%2C%22%2C-4sqrt%282%29%29%29\"\r\n" );
document.write( "and with the ellipse and get:\r\n" );
document.write( "\"A=%28matrix%281%2C3%2C-2%2C%22%2C%22%2C3sqrt%282%29%2F2%29%29\"  \r\n" );
document.write( "and you can tell by symmetry that \r\n" );
document.write( "\"B=%28matrix%281%2C3%2C-2%2C%22%2C%22%2C-3sqrt%282%29%2F2%29%29\"\r\n" );
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document.write( "The quadrilateral CDBA is a trapezoid (or trapezium if you live in the UK).\r\n" );
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document.write( "\"Area=expr%281%2F2%29%28b%5B1%5D%2Bb%5B2%5D%29h\"\r\n" );
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document.write( "\"b%5B1%5D=6sqrt%282%29%2F2=3sqrt%282%29\"\r\n" );
document.write( "\"b%5B2%5D=8sqrt%282%29\"\r\n" );
document.write( "The height is along the x-axis from -2 to 8 which is 10 units.\r\n" );
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document.write( "\"Area=expr%281%2F2%29%283sqrt%282%29%2B8sqrt%282%29%29%2A10=55sqrt%282%29\"\r\n" );
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document.write( "It's much easier If you've had calculus. Just equate the derivatives to find the\r\n" );
document.write( "values of x where the slopes are the same.  You'll get the same answer.\r\n" );
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document.write( "Edwin
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