document.write( "Question 1081620: Without graphing, Find the radius of a circle that is tangent to the y-axis that goes through two points (1,3), (2,4).
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Algebra.Com's Answer #695740 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The equation of a circle is\r\n" );
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document.write( "\"%28x-h%29%5E2%2B%28y-k%29%5E2=r%5E2\"\r\n" );
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document.write( "Since it's tangent to the y-axis, the center (h,k), is such that\r\n" );
document.write( "its x-coordinate h = ħr, but since the given points are in the\r\n" );
document.write( "first quadrant, the circle cannot go into the second quadrant and\r\n" );
document.write( "be tangent to the y-axis.  So h = r.\r\n" );
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document.write( "so we can substitute r for h\r\n" );
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document.write( "\"%28x-r%29%5E2%2B%28y-k%29%5E2=r%5E2\"\r\n" );
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document.write( "\"%28x-r%29%5E2-r%5E2=-%28y-k%29%5E2\"\r\n" );
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document.write( "Factor the left side as the difference of squares:\r\n" );
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document.write( "\"%28%28x-r%29%5E%22%22-r%29%28%28x-r%29%5E%22%22%2Br%29=-%28y-k%29%5E2\"\r\n" );
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document.write( "\"%28x-r-r%29%28x-r%2Br%29=-%28y%5E2-2ky%2Bk%5E2%29\"\r\n" );
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document.write( "\"%28x-2r%29%28x%29=-y%5E2%2B2ky%2Bk%5E2\"\r\n" );
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document.write( "\"x%5E2-2rx=-y%5E2%2B2ky%2Bk%5E2\"\r\n" );
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document.write( "\"x%5E2%2By%5E2-2rx-2ky-k%5E2=0\"\r\n" );
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document.write( "Substitute (x,y) = (1,3)\r\n" );
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document.write( "\"1%5E2%2B3%5E2-2r%281%29-2k%283%29-k%5E2=0\"\r\n" );
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document.write( "\"1%2B9-2r-6k-k%5E2=0\"\r\n" );
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document.write( "\"10-2r-6k-k%5E2=0\"\r\n" );
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document.write( "Solve for 2r, that's the variable I solved for.\r\n" );
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document.write( "{No need to solve for r, for that would induce denominators)\r\n" );
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document.write( "\"10-6k-k%5E2=2r\"\r\n" );
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document.write( "Substitute (x,y) = (2,4)\r\n" );
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document.write( "\"2%5E2%2B4%5E2-2r%282%29-2k%284%29-k%5E2=0\"\r\n" );
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document.write( "\"4%2B16-4r-8k-k%5E2=0\"\r\n" );
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document.write( "\"20-4r-8k-k%5E2=0\"\r\n" );
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document.write( "Solve for 4r\r\n" );
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document.write( "\"20-8k-k%5E2=4r\"\r\n" );
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document.write( "The system of equations to solve is\r\n" );
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document.write( "\"system%2810-6k-k%5E2=2r%2C20-8k-k%5E2=4r%29\"\r\n" );
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document.write( "Muliply both sides of the first equation by -2\r\n" );
document.write( "to eliminate the r-terms\r\n" );
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document.write( "\"system%28-20%2B12k%2B2k%5E2=-4r%2C20-8k-k%5E2=4r%29\"\r\n" );
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document.write( "Adding the two equations term by term:\r\n" );
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document.write( "\"4k-k%5E2=0\"\r\n" );
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document.write( "\"k%284-k%29=0\"\r\n" );
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document.write( "k = 0;  4-k = 0\r\n" );
document.write( "          4 = k\r\n" );
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document.write( "Using k = 0\r\n" );
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document.write( "\"10-6k-k%5E2=2r\"\r\n" );
document.write( "\"10-6%280%29-0%5E2=2r\"\r\n" );
document.write( "\"10=2r\"\r\n" );
document.write( "\"5=r\"\r\n" );
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document.write( "\"%28x-5%29%5E2%2B%28y-0%29%5E2=5%5E2\"\r\n" );
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document.write( "This is the graph of the circle with center (h,k)=(5,0) and radius r=5.\r\n" );
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document.write( "  \r\n" );
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document.write( "Using k = 4\r\n" );
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document.write( "\"10-6k-k%5E2=2r\"\r\n" );
document.write( "\"10-6%284%29-4%5E2=2r\"\r\n" );
document.write( "\"10-24-16=2r\"\r\n" );
document.write( "\"-30=2r\"\r\n" );
document.write( "\"-15=r\"\r\n" );
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document.write( "The radius cannot be negative.  So there is only one solution\r\n" );
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document.write( "Edwin
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