If the image of the point `(-3,k)` in the line `2x+y-2=0` is the point (1,5) then the value of k is (i) 2 (ii) 3 (iii) `-3` (iv) 1

Options

A. 2
B. 3
C. `-3`
D. 1

Answer

To solve the problem, we need to find the value of \( k \) such that the image of the point \( (-3, k) \) in the line \( 2x + y - 2 = 0 \) is the point \( (1, 5) \).

### Step-by-Step Solution:

1. **Identify the Points**:
Let \( A = (-3, k) \) be the original point and \( B = (1, 5) \) be the image of point \( A \).

2. **Find the Midpoint**:
The midpoint \( M \) of points \( A \) and \( B \) can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, \( x_1 = -3 \), \( y_1 = k \), \( x_2 = 1 \), and \( y_2 = 5 \).
So,
\[ M = \left( \frac{-3 + 1}{2}, \frac{k + 5}{2} \right) = \left( \frac{-2}{2}, \frac{k + 5}{2} \right) = (-1, \frac{k + 5}{2}) \]

3. **Substitute Midpoint into the Line Equation**:
The midpoint \( M \) must satisfy the line equation \( 2x + y - 2 = 0 \). Substituting \( M \) into the equation:
\[ 2(-1) + \frac{k + 5}{2} - 2 = 0 \]
Simplifying this:
\[ -2 + \frac{k + 5}{2} - 2 = 0 \]
\[ \frac{k + 5}{2} - 4 = 0 \]
\[ \frac{k + 5}{2} = 4 \]

4. **Solve for \( k \)**:
Multiply both sides by 2:
\[ k + 5 = 8 \]
Now, subtract 5 from both sides:
\[ k = 8 - 5 = 3 \]

5. **Conclusion**:
The value of \( k \) is \( 3 \).

### Final Answer:
The value of \( k \) is \( 3 \).