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2019 AMC 10B Problem 4

Problem 4 of 25EasierAlgebraGeometry

All lines with equation ax+by=cax+by=c such that a,a, b,b, cc form an arithmetic progression pass through a common point. What are the coordinates of that point?

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Solution

Let d=b−a.d=b-a. Then, we have (a,b,c)=(a,a+d,a+2d).(a,b,c)= (a,a+d,a+2d). Thus, ax+(a+d)y=a+2d.ax+(a+d)y=a+2d. If we match the parts of aa and d,d, we get ax+ay=aax+ay=a and dy=2ddy=2d for all a,d.a,d. Therefore, we have y=2,x+y=1y=2,x+y=1 implying that x=−1.x=-1. This makes the pair (−1,2).(-1,2). Thus, the answer is A .

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Concepts: linear equation · arithmetic sequence · coordinate geometry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.