Find the coordinates of a point P(x, y) lying on the line 2x + y=6 such that the sum of its distances from the fixed points A(1, 1) and B(4, 5) is minimized
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Step-by-Step Solution
Step 1: Understand the Problem and Reflection Principle
We need to find a point P on the line 2x+y=6 such that the sum of the distances from P to two fixed points A(1, 1) and B(4, 5) is minimized. This is a classic problem that can be solved using the reflection principle. The reflection principle states that the shortest path from point A to point B, touching a line L, is found by reflecting one of the points (say B) across the line L to get B', and then finding the straight line distance from A to B'. The intersection of AB' with line L will be the point P.
Step 2: Find the Reflection of Point B
Let B'(h, k) be the reflection of point B(4, 5) across the line 2x+y−6=0. The formula for the reflection of a point (x1,y1) across the line ax+by+c=0 is given by: ax−x1=by−y1=−2a2+b2ax1+by1+c Substituting the coordinates of B(4, 5) and the line equation 2x+y−6=0, we get the expression to find (h, k).
Step 3: Calculate Coordinates of B'
Now we calculate the value of the constant term and then solve for h and k. The constant term is −257=−514. Equating 2h−4 and 1k−5 to this value, we find the coordinates of B' to be (−58,511).
Step 4: Find the Equation of Line AB'
The point P that minimizes the sum of distances lies on the line segment connecting A and B'. We need to find the equation of the line passing through A(1, 1) and B'(−58,511). First, we calculate the slope of AB' using the formula m=x2−x1y2−y1. Then, we use the point-slope form of a linear equation, y−y1=m(x−x1), to find the equation of the line AB'.
Step 5: Find the Intersection Point P
The point P is the intersection of the given line 2x+y=6 and the line AB' (6x+13y=19). We can solve this system of linear equations to find the coordinates of P. We express y from the first equation and substitute it into the second equation to find x, and then find y.