document.write( "Question 1209658: The real numbers x and y satisfy
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Algebra.Com's Answer #849792 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "The real numbers x and y satisfy
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document.write( "Let x + y = k.    (1)\r\n" );
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document.write( "Then the quadratic equation represents a circle, while equation (1) represents a straight line.\r\n" );
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document.write( "So, the problem is to find the tangent line (1) to the circle with greatest k.\r\n" );
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document.write( "Or, in algebra language, we want to find k such a way, that quadratic equation and equation (1)\r\n" );
document.write( "have only one solution (representing the tangent point), which provides highest value of \"k\".\r\n" );
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document.write( "So, we express from (1)  y = k-x  and substitute it inte the quadratic equation.  We get then\r\n" );
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document.write( "    x^2 + (k-x)^2 - 8x + 6*(k-x) + 23 = 0,\r\n" );
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document.write( "    x^2 + k^2 - 2kx + x^2 - 8x + 6k - 6x + 23 = 0,\r\n" );
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document.write( "    2x^2 - (2k+14)x +(k^2 + 6k + 23) = 0.    (2)\r\n" );
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document.write( "The condition that the circle and the line have only one common point is equivalent \r\n" );
document.write( "to the condition that the discriminant of equation (2) is zero.\r\n" );
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document.write( "The discriminant is\r\n" );
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document.write( "    d = b^2 - 4ac = (2k+14)^2 - 4*2*(k^2 + 6k + 23) = 4k^2 + 56k + 106 - 8k^2 - 48k - 184 = \r\n" );
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document.write( "      = -4k^2 + 8k + 12.\r\n" );
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document.write( "Hence, the condition that the discriminant equal to zero is this quadratic equation for \"k\"\r\n" );
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document.write( "    4k^2 - 8k - 12 = 0.\r\n" );
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document.write( "Simplify and then factor\r\n" );
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document.write( "    k^2 - 2k - 3 = 0,\r\n" );
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document.write( "    (k-3)*(k+1) = 0.\r\n" );
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document.write( "The roots are k = 3  and k = -1.\r\n" );
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document.write( "We want the greatest \"k\", so the ANSWER to the problem's question is k = 3.\r\n" );
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