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AMC 10 USA Math Olympiad

Co-ordinate Geometry – AMC 10B – 2019 – Problem No – 4

The simplest example of power mean inequality is the arithmetic mean – geometric mean inequality. Learn in this self-learning module for math olympiad

Co-ordinate Geometry – AMC 10B – 2019 – Problem No – 4


This problem on co-ordinate geometry is from AMC 10B, 2019. First try it yourself.

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?

  • (-1,2)
  • (0,1)
  • (1,-2)
  • (1,0)

Key Concepts


Arithmetic Progression

2D – Co- ordinate Geometry

Cartesian System of Points

Check the Answer


Answer: (-1,2)

AMC 10B – 2019 – Problem No – 4

Challenges and Thrills in Pre-College Mathematics

Try with Hints


If all lines satisfy the condition, then we can just plug in values for a,b,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 =3

x+3y = 5 so

y= 2

Now plug the value of y into one of the previous equations. We get :

x+4=3

x = -1

Ans is (-1,2)

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