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document.write( "When k is a constant, find k for which the length of perpendicular dropped from point (2, 1) to line kx + y + 1 = 0 is √3.
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document.write( "Note: Can you please show your full solution? Thank you!
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document.write( "To make the solution easier, I re-formulate the problem in THIS EQUIVALENT WAY:\r
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document.write( " Find the coefficient \"k\" such that the line kx + y + 1 = 0 \r\n" );
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document.write( " is tangent to the circle of the radius 
centered at the point (2,1)\r\n" );
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document.write( " Solution\r
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document.write( "The standard form equation of the circle is\r\n" );
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+ 
= 3. (1)\r\n" );
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document.write( "The equation of the line is\r\n" );
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document.write( " y = -kx - 1. (2)\r\n" );
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document.write( "Substitute the equation (2) into equation (1), replacing y in the equation (1).\r\n" );
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document.write( "Doing this way, you will get then a single equation for only one unknown x\r\n" );
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document.write( " + 
= 3 (3)\r\n" );
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document.write( "or\r\n" );
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document.write( " + 
= 3. (4)\r\n" );
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document.write( "The idea of the solution to the problem is to reduce equation (4) to standard form quadratic equation\r\n" );
document.write( "and then to find the value of \"k\" from the condition that this equation has ONLY ONE real solution\r\n" );
document.write( "(which is equivalent to the fact that the line(2) and the circle (1) have ONLY ONE common point).\r\n" );
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document.write( "So, we simplify equation (4) step by step\r\n" );
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document.write( " x^2 - 4x + 4 + k^2x^2 + 4kx + 4 = 3\r\n" );
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document.write( " x^2(1+k^2) - 4x(1-k) + 5 = 0. (5)\r\n" );
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document.write( "Now we will find \"k\" from the condition that the discriminant of equation (5) is equal to zero.\r\n" );
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document.write( "The discriminant d is\r\n" );
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document.write( " d = b^2 - 4ac = 
- 
= 
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document.write( "The condition d = 0 is\r\n" );
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document.write( " = \r\n" );
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document.write( " 4(1-k)^2 = 5(1+k^2)\r\n" );
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document.write( " 4 - 8k + 4k^2 = 5 + 5k^2\r\n" );
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document.write( " k^2 + 8k + 1 = 0\r\n" );
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document.write( " 
= 
= 
= 
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document.write( "So, there are 2 (two, TWO) values for k: 
= 
and 
= 
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document.write( "ANSWER. There are two solutions for \"k\" : = and = .\r\n" );
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document.write( "Solved, answered and carefully explained in all details.\r
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document.write( "Do not forget to post your \"THANKS\" to me for my teaching.\r
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