document.write( "Question 1205414: If the solutions for the two equations below are the same, find the maximum value of k, if:\r
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Algebra.Com's Answer #842187 by Edwin McCravy(20060) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "That's a circle with center (-2,5) and radius 2 intersecting a line with\r\n" );
document.write( "slope 4 and y-intercept k. So, k, the y-intercept of the line will be the\r\n" );
document.write( "largest when the line is as far to the left of the circle and still intersects\r\n" );
document.write( "the circle.\r\n" );
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document.write( "That's when the line is tangent to the circle on the left side. That's also \r\n" );
document.write( "when the distance from the line to the center is the radius 2.\r\n" );
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document.write( "We use the point-to-line distance formula:\r\n" );
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document.write( "We get 0 on the right side of the line's equation:\r\n" );
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document.write( "Perpendicular distance from the point (x₁,y₁)\r\n" );
document.write( "to the line Ax+By+C=0 is\r\n" );
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document.write( "We want k to be a maximum so we use the + sign.\r\n" );
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document.write( " approximately 21.24621125.\r\n" );
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document.write( "Oh darn, I just realized that x and y had to be integers. Instead of\r\n" );
document.write( "starting over it looks like the nearest integer x could be -4. So we have\r\n" );
document.write( "to move the line a tiny bit right.\r\n" );
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document.write( "So, we plug -4 for x in the circle equation:\r\n" );
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document.write( "So (-4,5) is the nearest point to the point of tangency that\r\n" );
document.write( "has both coordinates integers.\r\n" );
document.write( "So the line y=4x+k should go through (-4,5), moving it a tiny\r\n" );
document.write( "bit right.\r\n" );
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document.write( "So the answer is 21.\r\n" );
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document.write( "Edwin

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