document.write( "Question 393102: Find the equation of the line normal to the curve \"y%5E2=9-24%2F%286-x%29\"
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Algebra.Com's Answer #279089 by Edwin McCravy(20060)\"\" \"About 
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Find the equation of the line normal to the curve \"y%5E2=9-24%2F%286-x%29\"
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document.write( "First we find the derivative of the curve using the implicit method:\r\n" );
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document.write( "\"y%5E2=9-24%2F%286-x%29\"\r\n" );
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document.write( "Rewrite that so that the denominator becomes a negative exponent \r\n" );
document.write( "in the top:\r\n" );
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document.write( "\"y%5E2=9-24%286-x%29%5E%28-1%29\"\r\n" );
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document.write( "\"2y%2Aexpr%28dy%2Fdx%29=0%2B24%286-x%29%5E%28-2%29%28-1%29\"\r\n" );
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document.write( "\"2y%2Aexpr%28dy%2Fdx%29=-24%286-x%29%5E%28-2%29\"\r\n" );
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document.write( "\"2y%2Aexpr%28dy%2Fdx%29=-24%2F%286-x%29%5E2\"\r\n" );
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document.write( "Divide both sides by 2:\r\n" );
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document.write( "\"y%2Aexpr%28dy%2Fdx%29=-12%2F%286-x%29%5E2\"\r\n" );
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document.write( "Divide both sides by y:\r\n" );
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document.write( "\"expr%28dy%2Fdx%29=-12%2F%28y%286-x%29%5E2%29\"\r\n" );
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document.write( "Substitute the point (3,1)\r\n" );
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document.write( "\"expr%28dy%2Fdx%29=-12%2F%28%281%29%286-%283%29%29%5E2%29\"\r\n" );
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document.write( "\"expr%28dy%2Fdx%29=-12%2F%283%29%5E2%29\"\r\n" );
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document.write( "\"expr%28dy%2Fdx%29=-12%2F9%29\"\r\n" );
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document.write( "\"expr%28dy%2Fdx%29=-4%2F3\"\r\n" );
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document.write( "So the slope of a tangent line to the graph at the point (3,1)\r\n" );
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document.write( "would be \"-4%2F3\"\r\n" );
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document.write( "Therefore the slope of a line normal (perpendicular) to the\r\n" );
document.write( "curve at that point has a slope which is the reciprocal of that\r\n" );
document.write( "slope with the opposite sign which would be \"%22%22%2B3%2F4\"\r\n" );
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document.write( "So we want the equation of the line with slope \"3%2F4\" that\r\n" );
document.write( "passes through (3,1)\r\n" );
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document.write( "\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"\r\n" );
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document.write( "\"y-1+=+expr%283%2F4%29%28x-3%29\"\r\n" );
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document.write( "Multiply through by 4\r\n" );
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document.write( "\"4y-4=3%28x-3%29\"\r\n" );
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document.write( "\"4y-4=3x-9\"\r\n" );
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document.write( "\"4y+=+3x-5\"\r\n" );
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document.write( "Divide through by 4\r\n" );
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document.write( "\"y+=+expr%283%2F4%29x-5%2F4\"\r\n" );
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document.write( "Now we graph that line in blue:\r\n" );
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document.write( "and we see that it looks perpendicular to the curve at the\r\n" );
document.write( "point (3,1).  Especially does it look perpendicular to the \r\n" );
document.write( "curve if we also draw the tangent line (in green):\r\n" );
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document.write( "Edwin

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