The mid-point of the line segment joining A(2a, 4) and B(-2, 3b) is (1, 2a +1). Find the value of a and b.

Answer

To find the values of \( a \) and \( b \), we will use the midpoint formula. The midpoint \( M \) of a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Given points:
- \( A(2a, 4) \)
- \( B(-2, 3b) \)
- Midpoint \( M(1, 2a + 1) \)

### Step 1: Set up the equations using the midpoint formula.

From the midpoint formula, we can equate the coordinates:

1. For the x-coordinate:
\[ \frac{2a + (-2)}{2} = 1 \]

2. For the y-coordinate:
\[ \frac{4 + 3b}{2} = 2a + 1 \]

### Step 2: Solve the x-coordinate equation.

Starting 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:
\[ a = 2 \]

### Step 3: Substitute \( a \) into the y-coordinate equation.

Now substitute \( a = 2 \) into the y-coordinate equation:
\[ \frac{4 + 3b}{2} = 2(2) + 1 \]

This simplifies to:
\[ \frac{4 + 3b}{2} = 4 + 1 \]
\[ \frac{4 + 3b}{2} = 5 \]

Multiply both sides by 2:
\[ 4 + 3b = 10 \]

Subtract 4 from both sides:
\[ 3b = 6 \]

Divide by 3:
\[ b = 2 \]

### Final Answer:
The values are:
\[ a = 2, \quad b = 2 \]