2019 AMC 10B Problems/Problem 4
Contents
Problem
All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates of that point?
Solution 1
If all lines satisfy the equation, then we can just plug in values for a, b, and c that form an arithmetic progression. Let's do a=1, b=2, c=3 and a=1, b=3, and c=5. Then the two lines we get are: Use elimination: Plug this into one of the previous lines. Thus the common point is
Solution 2
We know that are an arithmetic progression, so if the common difference is we can say Now we have , and expanding gives Factoring gives . Since this must always be true, we know that and , so and the common point is .
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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