2019 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 4 Easy

All lines with equation ax+by =c such that a,b,c form an arithmetic progression pass through a common point. What are the coordinates of that point? (2019 AMC 10B Problem, Question#4)

  • A.

    (-1,2)

  • B.

    (0,1)

  • C.

    (1,-2)

  • D.

    (1,0)

  • E.

    (1,2)

Answer:A

If all lines satisfy the condition, then we can just plug in values for a, b, and c that form an arithmetic progression. Let's use a = 1, b =2, c=3, and a =1, b=3, c=5.Then the two lines we get are: x+2y=3x+3y=5 Use elimination to deduce y =2 and plug this into one of the previous line equations. We get x+4 =3\Rightarrow x=-1 Thus the common point is \text {(A)}(-1,2).

We know that a, b, and c form an arithmetic progression, so if the common difference is d, we can say a, b, c=a, a+d, a+2d. Now we have ax + (a +d)y =a+2d, and expanding gives ax + ay +dy =a+2d. Factoring gives a(x +y-1) +d(y - 2) =0. Since this must always be true (regardless of the values of x and y), we must have x+y-1=0 and y-2=0, so x, y=-1, 2, and the common point is \text {(A)}(-1,2).

Related Problems
Problem 2 2019 AMC 10 B
Problem 3 2019 AMC 10 B
Problem 5 2019 AMC 10 B
Problem 6 2019 AMC 10 B
Think Academy Powered by ChatGPT