Key Concepts and Formulas

• Reflection Principle: The shortest distance between two points P and Q with a constraint of touching a line L is found by reflecting one of the points across the line L and then drawing a straight line from the other point to the reflected point. The intersection of this line with L gives the point of contact.
• Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. The square of the distance is $(x_2 - x_1)^2 + (y_2 - y_1)^2$.
• Equation of a Line: The equation of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ can be found using the point-slope form: $y - y_1 = m(x - x_1)$, where $m = \frac{y_2 - y_1}{x_2 - x_1}$ is the slope.

Step-by-Step Solution

1. Understand the Problem and Apply the Reflection Principle

We want to minimize the distance $PR + RQ$, where P is (-3, 4), Q is (0, 2), and R lies on the x-axis. By the reflection principle, we reflect Q across the x-axis to obtain Q'(0, -2). The minimum distance $PR + RQ$ is achieved when R lies on the line segment PQ'.

2. Find the Coordinates of the Reflected Point Q'

Reflecting the point Q(0, 2) across the x-axis means changing the sign of the y-coordinate. Therefore, Q' = (0, -2).

3. Find the Equation of the Line Passing Through P and Q'

We need to find the equation of the line passing through P(-3, 4) and Q'(0, -2).

• Calculate the slope ($m$): $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 4}{0 - (-3)} = \frac{-6}{3} = -2$
• Use the point-slope form with point Q'(0, -2): $y - (-2) = -2(x - 0)$ $y + 2 = -2x$ $2x + y + 2 = 0$

4. Find the Coordinates of Point R

Point R lies on the x-axis, so its y-coordinate is 0. Substitute y = 0 into the equation of the line PQ': $2x + 0 + 2 = 0$ $2x = -2$ $x = -1$ Thus, R = (-1, 0).

5. Calculate $PR^2$

We have P(-3, 4) and R(-1, 0). $PR^2 = (-1 - (-3))^2 + (0 - 4)^2$ $PR^2 = (-1 + 3)^2 + (-4)^2$ $PR^2 = (2)^2 + 16$ $PR^2 = 4 + 16 = 20$

6. Calculate $RQ^2$

We have R(-1, 0) and Q(0, 2). $RQ^2 = (0 - (-1))^2 + (2 - 0)^2$ $RQ^2 = (1)^2 + (2)^2$ $RQ^2 = 1 + 4 = 5$

7. Calculate the Final Expression

We need to find the value of $50(PR^2 + RQ^2)$. $50(PR^2 + RQ^2) = 50(20 + 5)$ $= 50(25)$ $= 1250$

Common Mistakes & Tips

• Reflection Across X-axis: Make sure to reflect the point correctly across the x-axis, which means changing the sign of the y-coordinate only.
• Calculating the Correct Expression: The problem asks for $50(PR^2 + RQ^2)$, not $50(PQ'^2)$. Calculate $PR^2$ and $RQ^2$ separately.
• Sign Errors: Be careful with signs when calculating the slope and using the distance formula.

Summary

To minimize the distance traveled from P to Q touching the x-axis, we reflect Q across the x-axis to Q'. The point R lies on the line segment PQ'. By finding the equation of the line PQ' and setting y=0, we find R(-1, 0). Then we calculate $PR^2 = 20$ and $RQ^2 = 5$, so $50(PR^2 + RQ^2) = 50(20 + 5) = 1250$.

The final answer is \boxed{1250}.