A = (−2,4) and B = (7,−2). Find the point Q

Question

Let A = (−2,4) and B = (7,−2). Find the point Q on the line y = 2 that makes the total distance AQ + BQ as small as possible.

Answer 1

The hardest part about this problem is understanding where to start. To help explain, I've added some steps. First, the shortest way to get from point A to point B is along a straight line. That means the shortest distance for AQ + BQ is when Q is on the line from A to B. So we should first find the line between A and B.

m = (y₁-y₂)/(x₁-x₂) = (4-(-2))/(-2-7) = -2/3

So

y = -2/3x + b

Solving for b using point A

y - 4 = -2/3(x + 2)

y = -2/3x + 8/3

We know that point Q lies on this line. We also know that point Q's y coordinate is 2. So, solving for x:

2 = -2/3x + 8/3

-2/3 = -2/3x

x = 1

That means that point Q is located at (1, 2).