document.write( "Question 1171717: What is the maximum value of k, if the solutions for the two equations below are the same, and both x and y are integers?\r
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Algebra.Com's Answer #796711 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"system%28%28x%2B5%29%5E2+%2B+%28y-1%29%5E2+=+25%2C-2x+%2B+3y+=+k%29\"\r\n" );
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document.write( "Put the second in slope-intercept form:\r\n" );
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document.write( "\"-2x+%2B+3y+=+k\"\r\n" );
document.write( "\"3y+=+2x%2Bk\"\r\n" );
document.write( "\"y+=+expr%282%2F3%29x%2Bk%2F3\"\r\n" );
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document.write( "The first equation is a circle and the second represents a set of parallel lines with slope 2/3 and y-intercept k/3.\r\n" );
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document.write( "We want to find the maximum value of k.  k will be a maximum when k/3 is\r\n" );
document.write( "a maximum. k/3 is the y-intercept and it will be a maximum when the y-intercept\r\n" );
document.write( "is the largest.  That will be when the line is the upper tangent to the circle.\r\n" );
document.write( "Like this:\r\n" );
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document.write( "We know the distance from the center (-5,1) to the point of tangency is the\r\n" );
document.write( "radius, which is 5. So we use the formula for the perpendicular distance\r\n" );
document.write( "from a point to a line, which is\r\n" );
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document.write( "The perpendicular distance from the point (x1,y1) to the line Ax + By + C = 0\r\n" );
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document.write( "is given by the formula: \r\n" );
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document.write( "\"d=abs%28Ax%5B1%5D%2BBy%5B1%5D%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29\"\r\n" );
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document.write( "So the perpendicular distance from the center (-5,1) to the line\r\n" );
document.write( "-2x + 3y - k = 0 is equal to the radius 5, so\r\n" );
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document.write( "\"abs%28-2%28-5%29%2B3%281%29-k%29%2Fsqrt%28%28-2%29%5E2%2B%283%29%5E2%29=5\"\r\n" );
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document.write( "\"abs%2810%2B3-k%29%2Fsqrt%284%2B9%29=5\"\r\n" );
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document.write( "\"abs%2813-k%29%2Fsqrt%2813%29=5\"\r\n" );
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document.write( "\"abs%2813-k%29=5sqrt%2813%29\"\r\n" );
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document.write( "\"13-k=5sqrt%2813%29\" or \"-13%2Bk=5sqrt%2813%29\"\r\n" );
document.write( "\"-k=-13%2B5sqrt%2813%29\" or \"k=13%2B5sqrt%2813%29\"\r\n" );
document.write( "\"k=13-5sqrt%2813%29\" \r\n" );
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document.write( "The first is the lower tangent line, so we\r\n" );
document.write( "want the upper tangent line,\r\n" );
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document.write( "So the maximum value of k is\r\n" );
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document.write( "\"k=13%2B5sqrt%2813%29\" which is approximately 31.02775638\r\n" );
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document.write( "The largest potential integer k is 31.\r\n" );
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document.write( "And it does turn out that when k=31\r\n" );
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document.write( "\"system%28%28x%2B5%29%5E2+%2B+%28y-1%29%5E2+=+25%2C-2x+%2B+3y+=+31%29\"\r\n" );
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document.write( "We get two solutions (x,y) = (-8,5) and (x,y) = \"%28matrix%281%2C3%2C-98%2F13%2C%22%2C%22%2C69%2F13%29%29\"\r\n" );
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document.write( "So x=-8, y=5, is an integer solution that allows k to be the maximum integer\r\n" );
document.write( "value of 31. \r\n" );
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document.write( "Edwin
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