# Maximal distance between points on a line

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?

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.

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

• What if the line does not pass through $(0,0)$? – fahrbach May 27 '14 at 16:51
• 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. – David K May 27 '14 at 16:53