The mid-point of the line segment joining (2a, 4) and (-2, 2b) is (1, 2a + 1). The values of a and b are 

Options

A. a = 3, b = -1
B. a = 2,b = -3
C. a = 3, b = -2
D. a = 2, b = 3

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given that the mid-point of the line segment joining the points \( (2a, 4) \) and \( (-2, 2b) \) is \( (1, 2a + 1) \).

### Step-by-Step Solution:

1. **Understand the Midpoint Formula**:
The midpoint \( M \) of a line segment joining points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

2. **Identify the Coordinates**:
Here, the coordinates are:
- Point 1: \( (x_1, y_1) = (2a, 4) \)
- Point 2: \( (x_2, y_2) = (-2, 2b) \)
- Midpoint: \( (1, 2a + 1) \)

3. **Set Up the Equations for the Midpoint**:
Using the midpoint formula, we can set up two equations:
- For the x-coordinate:
\[ \frac{2a + (-2)}{2} = 1 \]
- For the y-coordinate:
\[ \frac{4 + 2b}{2} = 2a + 1 \]

4. **Solve the x-coordinate Equation**:
Start with the x-coordinate equation:
\[ \frac{2a - 2}{2} = 1 \]
Multiply both sides by 2:
\[ 2a - 2 = 2 \]
Add 2 to both sides:
\[ 2a = 4 \]
Divide by 2:
\[ a = 2 \]

5. **Substitute \( a \) into the y-coordinate Equation**:
Now substitute \( a = 2 \) into the y-coordinate equation:
\[ \frac{4 + 2b}{2} = 2(2) + 1 \]
Simplify the right side:
\[ \frac{4 + 2b}{2} = 4 + 1 = 5 \]
Multiply both sides by 2:
\[ 4 + 2b = 10 \]
Subtract 4 from both sides:
\[ 2b = 6 \]
Divide by 2:
\[ b = 3 \]

6. **Final Values**:
Therefore, the values of \( a \) and \( b \) are:
\[ a = 2, \quad b = 3 \]

### Summary:
The values of \( a \) and \( b \) are \( 2 \) and \( 3 \), respectively.