document.write( "Question 1066886: Find the equation of the tangent to the curve (x+2y)^2+2x-y-3=0 parallel the line 4x+3y=12. \n" ); document.write( "
Algebra.Com's Answer #682198 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The red curve is the graph of the equation \"%28x%2B2y%29%5E2%2B2x-y-3=0\"\r\n" );
document.write( "The black line is the graph of the equation \"4x%2B3y=12\" \r\n" );
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document.write( "Solve \"4x%2B3y=12\" for y:\r\n" );
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document.write( "       \"y=expr%28-4%2F3%29x%2B4\"\r\n" );
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document.write( "The slope of any line parallel to it has slope \"-4%2F3\"\r\n" );
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document.write( "and the equation of any line parallel to \"y=expr%28-4%2F3%29x%2B4\"\r\n" );
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document.write( "has equation \"y=expr%28-4%2F3%29x%2Bb\" for some value of b\r\n" );
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document.write( "So we solve the system by substitution: \r\n" );
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document.write( "\"system%28%28x%2B2y%29%5E2%2B2x-y-3=0%2Cy=expr%28-4%2F3%29x%2Bb%29\"\r\n" );
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document.write( "You do that and solve for x.  I did it on paper, but I don't \r\n" );
document.write( "feel like typing all the steps in.  If you can't get it, tell \r\n" );
document.write( "me in the thank-you note form below and I'll help you do it.\r\n" );
document.write( "It involves the quadratic formula.\r\n" );
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document.write( "Here is what I got for the x values of the two solutions:\r\n" );
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document.write( "\"x=expr%283%2F5%29%28b-1+%2B-+sqrt%284-3b%29%29\"\r\n" );
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document.write( "The line will be tangent to the curve if both solutions are\r\n" );
document.write( "equal.  We see that they will be equal if the square roots are \r\n" );
document.write( "both 0.\r\n" );
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document.write( "So we set what's under the square root equal to 0:\r\n" );
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document.write( "\"4-3b=0\"\r\n" );
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document.write( "\"b=+4%2F3\"\r\n" );
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document.write( "So the equation of the line tangent to the curve parallel to\r\n" );
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document.write( "4x+3y=12\r\n" );
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document.write( "is   \"y=expr%28-4%2F3%29x%2B4%2F3\"\r\n" );
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document.write( "which in general form is\r\n" );
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document.write( "\"4x%2B3y=4\"\r\n" );
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document.write( "We draw it in green:\r\n" );
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document.write( "It turns out that the line is tangent at the point (1,0).\r\n" );
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document.write( "Edwin
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