document.write( "Question 1024669: find the equation of the line tangent to the ellipse x^2+3y^2-x+2y=0 at the origin. Thank you! \n" ); document.write( "
Algebra.Com's Answer #640061 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!

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This can be done with or without calculus.  I'll do it\r\n" );
document.write( "both ways since I don't know what math you are taking:\r\n" );
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document.write( "Without using calculus:\r\n" );
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document.write( "Any line through the origin has slope m and y-intercept\r\n" );
document.write( "(0,0), so its equation is y = mx\r\n" );
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document.write( "If a line is tangent to a curve, the two solutions to\r\n" );
document.write( "the system of equations of the line and the curve\r\n" );
document.write( "will be equal.\r\n" );
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document.write( "So we solve the system:\r\n" );
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document.write( "\"system%28x%5E2%2B3y%5E2-x%2B2y=0%2Cy=mx%29\"\r\n" );
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document.write( "by substitution and set the discriminant = 0,\r\n" );
document.write( "so that the two solutions will be the same.\r\n" );
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document.write( "\"x%5E2%2B3y%5E2-x%2B2y=0\"\r\n" );
document.write( "\"x%5E2%2B3%28mx%29%5E2-x%2B2%28mx%29=0\"\r\n" );
document.write( "\"x%5E2%2B3m%5E2x%5E2-x%2B2mx=0\"\r\n" );
document.write( "\"%281%2B3m%5E2%29x%5E2%2B%28-1%2B2m%29x=0\"\r\n" );
document.write( "\"%281%2B3m%5E2%29x%5E2%2B%28-1%2B2m%29x=0\"\r\n" );
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document.write( "The discriminant = \"b%5E2-4ac\" = \r\n" );
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document.write( "\"%28-1%2B2m%29%5E2-4%28%281%2B3m%5E2%29%2A0%29\" =\r\n" );
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document.write( "\"%28-1%2B2m%29%5E2\" =\r\n" );
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document.write( "Set that = 0:\r\n" );
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document.write( "\"%28-1%2B2m%29%5E2=0\" =\r\n" );
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document.write( "\"-1%2B2m=0\"\r\n" );
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document.write( "\"2m=1\"\r\n" );
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document.write( "\"m=1%2F2\"\r\n" );
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document.write( "So slope = m = \"1%2F2\"\r\n" );
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document.write( "Since the y-intercept is (0,0)\r\n" );
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document.write( "\"y+=+mx+%2B+b\"  becomes\r\n" );
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document.write( "\"y=expr%281%2F2%29x\"\r\n" );
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document.write( "------------------------\r\n" );
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document.write( "Using calculus:\r\n" );
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document.write( "\"x%5E2%2B3y%5E2-x%2B2y=0\"\r\n" );
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document.write( "Differentiate implicitly:\r\n" );
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document.write( "\"2x%2B6y%2A%22y%27%22-1%2B2%2A%22y%27%22=0\" \r\n" );
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document.write( "Substitute x=0, y=0\r\n" );
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document.write( "\"2%280%29%2B6%280%29%2A%22y%27%22-1%2B2%2A%22y%27%22=0\"\r\n" );
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document.write( "\"-1%2B2%2A%22y%27%22=0\"\r\n" );
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document.write( "\"2%2A%22y%27%22=1\"\r\n" );
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document.write( "\"%22y%27%22=1%2F2\"\r\n" );
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document.write( "So slope = m = \"1%2F2\"\r\n" );
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document.write( "Since the y-intercept is (0,0)\r\n" );
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document.write( "\"y+=+mx+%2B+b\"  becomes\r\n" );
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document.write( "\"y=expr%281%2F2%29x\"\r\n" );
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document.write( "Edwin
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