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

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 $(x_1, y_1)$ across the line $ax + by + c = 0$ is given by: $$\frac{x - x_1}{a} = \frac{y - y_1}{b} = -2 \frac{ax_1 + by_1 + c}{a^2 + b^2}$$ 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 $-2 \frac{7}{5} = -\frac{14}{5}$. Equating $\frac{h-4}{2}$ and $\frac{k-5}{1}$ to this value, we find the coordinates of B' to be $(-\frac{8}{5}, \frac{11}{5})$.

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'($-\frac{8}{5}, \frac{11}{5}$). First, we calculate the slope of AB' using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Then, we use the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, 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 $y$.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

All Coordinate Geometry solutions →