document.write( "Question 1184256: Find the values of k for which the line y=kx-10 meets the curve x^2+y^2=10x. \n" ); document.write( "
Algebra.Com's Answer #815028 by ikleyn(52786)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            In her post,  @MathLover1  INCORRECTLY  interprets the problem,\r
\n" ); document.write( "\n" ); document.write( "            considering only the case when the straight line touches the circle.\r
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\n" ); document.write( "\n" ); document.write( "            The correct interpretation is to consider a general case of an intersection
\n" ); document.write( "            (\"having at least one intersection point\").\r
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\n" ); document.write( "\n" ); document.write( "            So,  I  came to bring a  CORRECT  SOLUTION.\r
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document.write( "\"x%5E2%2By%5E2=10x\"....eq.1\r\n" );
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document.write( "\"y=kx-10+\".....eq.2\r\n" );
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document.write( "substitute \"y\" in eq.1\r\n" );
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document.write( "\"x%5E2%2B%28kx-10%29%5E2+=+10x\"\r\n" );
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document.write( "\"x%5E2%2Bk%5E2x%5E2+-+20kx+-10x%2B+100=0\"\r\n" );
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document.write( "\"%281%2Bk%5E2%29x%5E2+-+%2820k+%2B10%29x%2B+100=0\"\r\n" );
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document.write( "if the line meets the curve, there is at least one solution and it is a case if discriminant >= \"0\"\r\n" );
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document.write( "so, use discriminant \r\n" );
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document.write( "\"b%5E2-4ac+%3E=+0\"  ...in your case \"a=%281%2Bk%5E2%29\", \"b=-+%2820k+%2B10%29\", and \"c=100\"\r\n" );
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document.write( "\"%28-+%2820k+%2B10%29%29%5E2-4%281%2Bk%5E2%29%2A100+%3E=+0++\"\r\n" );
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document.write( "\"400k%5E2+%2B+400k+%2B+100-400+k%5E2+-+400+%3E=+0+\"\r\n" );
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document.write( " \"400k+-+300+%3E=+0++\"\r\n" );
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document.write( "\"+400k+%3E=+300+\"\r\n" );
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document.write( "\"4k+%3E=+3\"\r\n" );
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document.write( "\"k+%3E=+3%2F4\"\r\n" );
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document.write( "ANSWER.  k >= \"3%2F4\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved (correctly).\r
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