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Two points A and B are on different sides of a line. Find a point Y on the line such that the absolute value of the difference from Y to A and Y to B is maximal.

My thoughts are as follows. Let's say that point Y is on a line but is not collinear with A and B. Then I could draw a triangle by joining the 3 points. For this triangle, segment AB is less than the sum of AY and BY. Then AB is greater than the absolute value of (YA-YB). How can I prove that this difference is maximal?

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Hint: Find the function that yields the difference in the distances from $A$ to $Y$ and $B$ to $Y$ using the distance formula. This function will be of one variable since we know $A$, $B$, and the equation for the line. Then find the minimum or maximum of this function since we're looking for the greatest absolute value.

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(0,0) because that would make both lines the maximum size they could be

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    $\begingroup$ What if the line does not pass through $(0,0)$? $\endgroup$ – fahrbach May 27 '14 at 16:51
  • $\begingroup$ And even if it does, the two points A and B could be anywhere. If (0,0) is the correct point for one value of A and B, just move them over to the right and now (0,0) is no longer correct. $\endgroup$ – David K May 27 '14 at 16:53

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