2019 AMC 10B Problems/Problem 4
Problem
All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates of that point?
Solution
Solution 1
If all lines satisfy the condition, then we can just plug in values for , , and that form an arithmetic progression. Let's use , , , and , , . Then the two lines we get are: Use elimination to deduce and plug this into one of the previous line equations. We get Thus the common point is .
~IronicNinja
Solution 2
We know that , , and form 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 (regardless of the values of and ), we must have and , so and the common point is .
Solution 3
We use process of elimination. doesn't necessarily work because isn't always true. also doesn't necessarily work because the x-value is , but the y-value is an integer. So by process of elimination, is our answer. ~Baolan
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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