If the image of point `P(2, 3)` in a line `L` is `Q (4,5)` then, the image of point `R (0,0)` in the same line is:

Options

A. (3,2)
B. (-2,3)
C. (-3,-2)
D. (3,-2)

Explanation

To find the image of point \( R(0,0) \) in the same line \( L \) where the image of point \( P(2,3) \) is \( Q(4,5) \), we can follow these steps:

### Step 1: Find the Midpoint of Points P and Q
The midpoint \( M \) of points \( P(2,3) \) and \( Q(4,5) \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Substituting the coordinates of \( P \) and \( Q \):
\[ M = \left( \frac{2 + 4}{2}, \frac{3 + 5}{2} \right) = \left( \frac{6}{2}, \frac{8}{2} \right) = (3, 4) \]

### Step 2: Determine the Slope of Line PQ
The slope \( m_{PQ} \) of line \( PQ \) can be calculated using the formula:
\[ m_{PQ} = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of \( P \) and \( Q \):
\[ m_{PQ} = \frac{5 - 3}{4 - 2} = \frac{2}{2} = 1 \]

### Step 3: Find the Slope of Line L
Since line \( L \) is perpendicular to line \( PQ \), we can find the slope of line \( L \) using the relationship:
\[ m_{PQ} \cdot m_{L} = -1 \]
Thus,
\[ 1 \cdot m_{L} = -1 \implies m_{L} = -1 \]

### Step 4: Write the Equation of Line L
Using the point-slope form of the line equation:
\[ y - y_1 = m(x - x_1) \]
where \( (x_1, y_1) = (3, 4) \) and \( m = -1 \):
\[ y - 4 = -1(x - 3) \]
Simplifying this:
\[ y - 4 = -x + 3 \implies x + y - 7 = 0 \]

### Step 5: Find the Image of Point R(0,0)
Let the image of point \( R(0,0) \) be \( S(p,q) \). The midpoint \( N \) of \( R(0,0) \) and \( S(p,q) \) must lie on line \( L \):
\[ N = \left( \frac{0 + p}{2}, \frac{0 + q}{2} \right) \]
Substituting into the line equation:
\[ \frac{p}{2} + \frac{q}{2} - 7 = 0 \implies p + q = 14 \quad \text{(1)} \]

### Step 6: Use the Slope Condition
The slope of line \( RS \) is given by:
\[ \text{slope of } RS = \frac{q - 0}{p - 0} = \frac{q}{p} \]
Since line \( RS \) is perpendicular to line \( L \) (which has a slope of -1):
\[ \frac{q}{p} \cdot (-1) = -1 \implies q = p \quad \text{(2)} \]

### Step 7: Solve the System of Equations
From equations (1) and (2):
1. \( p + q = 14 \)
2. \( q = p \)

Substituting \( q = p \) into equation (1):
\[ p + p = 14 \implies 2p = 14 \implies p = 7 \]
Thus, \( q = 7 \).

### Conclusion
The image of point \( R(0,0) \) in line \( L \) is:
\[ S(7,7) \]