document.write( "Question 1132758: The circle x^2 + (y-c)^2=r^2, where c >0 and r>0, lies inside the parabola y=x^2. The circle touches the parabola at exactly two points located symmetrically on opposite sides of the y-axis, as shown in the diagram. Deduce that c>1/2\r
\n" ); document.write( "\n" ); document.write( "so in the diagram, the parabola has vertex (0,0) and is positive for all values of x, and (o,c) is above the two points of intersection.\r
\n" ); document.write( "\n" ); document.write( "this was 2012 Q16c) HSC question \n" ); document.write( "
Algebra.Com's Answer #749897 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "First we find the points of intersection of the circle and\r\n" );
document.write( "parabola by solving the system:\r\n" );
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document.write( "\"system%28x%5E2+%2B+%28y-c%29%5E2=r%5E2%2Cy=x%5E2%29\" \r\n" );
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document.write( "We get this solution:\r\n" );
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document.write( "\"y=x%5E2+=+%22%22+%2B-+expr%281%2F2%29sqrt%284r%5E2+-4c+%2B+1%29+%2B+c+-+1%2F2%29\"\r\n" );
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document.write( "Since the circle touches the parabola in 2 places, like this:\r\n" );
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document.write( "the value of c must be greater than it is in the case \r\n" );
document.write( "where all four solutions coincide and are equal at the origin (0,0). \r\n" );
document.write( "That is the case when y=0, and r=c\r\n" );
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document.write( "\"y=x%5E2+=+%22%22+%2B-+expr%281%2F2%29sqrt%284r%5E2+-4c+%2B+1%29+%2B+c+-+1%2F2%29\"\r\n" );
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document.write( "To find c in that case, we setting y=0 and r=c\r\n" );
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document.write( "\"0=+expr%281%2F2%29sqrt%284c%5E2+-4c+%2B+1%29+%2B+c+-+1%2F2%29\"\r\n" );
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document.write( "\"0=+expr%281%2F2%29sqrt%28%282c-1%29%5E2%29+%2B+c+-+1%2F2%29\"\r\n" );
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document.write( "\"0=+sqrt%28%282c-1%29%5E2%29+%2B+2c+-+1%29\"\r\n" );
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document.write( "\"0=+%282c-1%29+%2B+2c+-+1%29\"\r\n" );
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document.write( "\"0=+2c-1+%2B+2c+-+1%29\"\r\n" );
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document.write( "\"0=+4c+-+2%29\"\r\n" );
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document.write( "\"2+=+4c\"\r\n" );
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document.write( "\"2%2F4=c\"\r\n" );
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document.write( "\"1%2F2=c\"\r\n" );
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document.write( "That is the case when c=r=1/2, therefore we must have c>1/2.\r\n" );
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document.write( "Edwin
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