If P, Q and R are three points with coordinates `(1, 4), (4, 5) and (m, m)` respectively, then the value of m for which `PR + RQ` is minimum, is :

Options

A. 4
B. 3
C. `(17)/(8)`
D. `(7)/(2)`

Answer

To find the value of \( m \) for which \( PR + RQ \) is minimum, we can follow these steps:

### Step 1: Identify the points
The coordinates of the points are given as:
- \( P(1, 4) \)
- \( Q(4, 5) \)
- \( R(m, m) \)

### Step 2: Understand the geometric interpretation
We need to minimize the distance \( PR + RQ \). To do this effectively, we can use the concept of the mirror image. We will find the mirror image of point \( Q \) with respect to the line \( y = x \).

### Step 3: Find the mirror image of point \( Q \)
The coordinates of point \( Q \) are \( (4, 5) \). The mirror image \( Q' \) of point \( Q \) across the line \( y = x \) will have coordinates \( (5, 4) \).

### Step 4: Set up the condition for collinearity
For the points \( P, R, \) and \( Q' \) to be collinear, the slopes of the lines \( PR \) and \( RQ' \) must be equal.

The slope of line \( PR \) is given by:
\[ \text{slope of } PR = \frac{m - 4}{m - 1} \]

The slope of line \( RQ' \) is given by:
\[ \text{slope of } RQ' = \frac{4 - m}{5 - m} \]

### Step 5: Set the slopes equal
Setting the slopes equal gives us:
\[ \frac{m - 4}{m - 1} = \frac{4 - m}{5 - m} \]

### Step 6: Cross-multiply to solve for \( m \)
Cross-multiplying yields:
\[ (m - 4)(5 - m) = (4 - m)(m - 1) \]

Expanding both sides:
\[ 5m - m^2 - 20 + 4m = 4m - 4 - m^2 + m \]

### Step 7: Simplify the equation
This simplifies to:
\[ 9m - 20 = 5m - 4 \]
\[ 9m - 5m = 20 - 4 \]
\[ 4m = 16 \]
\[ m = 4 \]

### Conclusion
The value of \( m \) for which \( PR + RQ \) is minimum is \( m = 4 \).