Problem:
All lines with equation such that form an arithmetic progression pass through a common point. What are the coordinates of that point?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If form an arithmetic progression, then and for some number . Then the given linear equation becomes , which is equivalent to
This will hold for all values of and if and only if and , which means and . Thus is the unique point through which all such lines pass.
If form an arithmetic progression, then . This equation is equivalent to , so and give a point through which the line passes. Conversely, if , and , then the equation of the line is , and none of the other four choices satisfies this equation.
Letting yields , and letting yields . In fact, is on the line if and only if , which is equivalent to , which is the defining condition for to be an arithmetic progression.
The problems on this page are the property of the MAA's American Mathematics Competitions