document.write( "Question 829382: Find the equation of normal to the curve y= x sqrt(1-2x) at the point (-4,-12). \n" ); document.write( "
Algebra.Com's Answer #499953 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "y = \"x%2Asqrt%281-2x%29\" at the point (-4,-12)\r\n" );
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document.write( "The derivative is the slope of the tangent line.\r\n" );
document.write( "The slope of the normal is the negative reciprocal of\r\n" );
document.write( "the slope of the tangent line.\r\n" );
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document.write( "So.\r\n" );
document.write( "1. We find the derivative.\r\n" );
document.write( "2. We substitute x=-4 and simplify\r\n" );
document.write( "3. We take the negative reciprocal of the result of 2\r\n" );
document.write( "4. We use the point-slope formula with this slope and the given point.\r\n" );
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document.write( "\"y\"\"%22%22=%22%22\"\"x%2Asqrt%281-2x%29\"\r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Cy%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2Cx%2A%281-2x%29%5E%281%2F2%29%29\" \r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Cdy%2Fdx%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2Cx%2A%281%2F2%29%281-2x%29%5E%28-1%2F2%29%28-2%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%281-2x%29%5E%281%2F2%29%281%29%29\" \r\n" );
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document.write( "\"matrix%282%2C1%2C%22%22%2Cdy%2Fdx%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C-x%281-2x%29%5E%28-1%2F2%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%281-2x%29%5E%281%2F2%29%29\" \r\n" );
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document.write( "We substitute x=-4\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C-%28-4%29%281-2%28-4%29%29%5E%28-1%2F2%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%281-2%28-4%29%29%5E%281%2F2%29%29\"\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C4%281%2B8%29%5E%28-1%2F2%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%281%2B8%29%5E%281%2F2%29%29\"\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C4%289%29%5E%28-1%2F2%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%289%29%5E%281%2F2%29%29\"\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C4%289%29%5E%28-1%2F2%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C%289%29%5E%281%2F2%29%29\"\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C4%2Fsqrt%289%29%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2Csqrt%289%29%29\"\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C4%2F3%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C3%29\"\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C4%2F3%29\"\"matrix%282%2C1%2C%22%22%2C%22%22%2B%22%22%29\"\"matrix%282%2C1%2C%22%22%2C9%2F3%29\"\r\n" );
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document.write( "\"matrix%282%2C2%2C%22%22%2C%22%22%2Cdy%2Fdx%2Cat_x=-4%29\"\"matrix%282%2C1%2C%22%22%2C%22%22=%22%22%29\"\"matrix%282%2C1%2C%22%22%2C13%2F3%29\"\r\n" );
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document.write( "The slope of the tangent line is \"13%2F3\", so the slope of the\r\n" );
document.write( "normal is \"-3%2F13\"\r\n" );
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document.write( "So we want the equation of the line through (-4,-12) with slope \"-3%2F13\"\r\n" );
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document.write( "y - y1 = m(x - x1)\r\n" );
document.write( "where (x1,y1) = (-4,-12)\r\n" );
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document.write( "\"y+-+%28-12%29\"\"%22%22=%22%22\"\"expr%28-3%2F13%29%28x-%28-4%29%29\"\r\n" );
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document.write( "\"y+%2B12%29\"\"%22%22=%22%22\"\"expr%28-3%2F13%29%28x%2B4%29\"\r\n" );
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document.write( "Multiply through by 13\r\n" );
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document.write( "13y+156 = -3(x+4)\r\n" );
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document.write( "13y+156 = -3x-12\r\n" );
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document.write( " 3x+13y = -168\r\n" );
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document.write( "Here's the graph, it looks like a normal (perpendicular) line.\r\n" );
document.write( "So it must be correct.\r\n" );
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document.write( "Edwin
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