document.write( "Question 1120571: For what values of m is the line y= mx a tangent to the parabola y=x^2 - 8x + 25? \n" ); document.write( "

Algebra.Com's Answer #736239 by ikleyn(52835) $\"\"$ $\"About$

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document.write( "1.  The equation for the common point of the straight line  y = mx  and the parabola  y = x^2 - 8x + 25  is\r\n" );
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document.write( "    mx = x^2 - 8x + 25,    or\r\n" );
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document.write( "    x^2 - (8+m)*x + 25 = 0.        (1)\r\n" );
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document.write( "2.  The straight line is the tangent to the parabola if the two roots of the equation (1) merge into one root.\r\n" );
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document.write( "    The necessary and sufficient condition for it is equality of the discriminant of the equation (1) to zero:\r\n" );
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document.write( "    d = b^2 - 4ac = 0,\r\n" );
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document.write( "        where  a= 1, b = -(8+m)  and  c= 25:\r\n" );
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document.write( "    d =  = 0,    or\r\n" );
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document.write( "    64 + 16m + m^2 - 100 = 0,\r\n" );
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document.write( "    m^2 + 16m - 36 = 0,\r\n" );
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document.write( "    (m-2)*(m+18) = 0.\r\n" );
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document.write( "This equation has two solutions:  m= 2  and  m= -18.\r\n" );
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document.write( "Answer.  The problem has two solutions:  m= 2  and  m= -18.\r\n" );
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