If the points `A(-2,-1),B(1,0),C(a,3)` and `D(1,b)` form a parallelogram ABCD, Then the values of a and b are

Options

A. `a=-4,b=-2`
B. `a=-4,b=2`
C. `a=4,b=2`
D. `a=2,b=-4`

Answer

To solve the problem of finding the values of \( a \) and \( b \) such that the points \( A(-2,-1) \), \( B(1,0) \), \( C(a,3) \), and \( D(1,b) \) form a parallelogram, we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-step Solution:

1. **Identify the midpoints of the diagonals**:
The diagonals of the parallelogram are \( AC \) and \( BD \). The midpoints of these diagonals must be equal.

2. **Calculate the midpoint of diagonal \( AC \)**:
The coordinates of points \( A \) and \( C \) are \( A(-2, -1) \) and \( C(a, 3) \).
The midpoint \( M_{AC} \) is given by:
\[ M_{AC} = \left( \frac{-2 + a}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{-2 + a}{2}, 1 \right) \]

3. **Calculate the midpoint of diagonal \( BD \)**:
The coordinates of points \( B \) and \( D \) are \( B(1, 0) \) and \( D(1, b) \).
The midpoint \( M_{BD} \) is given by:
\[ M_{BD} = \left( \frac{1 + 1}{2}, \frac{0 + b}{2} \right) = \left( 1, \frac{b}{2} \right) \]

4. **Set the midpoints equal to each other**:
Since \( M_{AC} = M_{BD} \), we can set the coordinates equal:
\[ \frac{-2 + a}{2} = 1 \quad \text{(1)} \]
\[ 1 = \frac{b}{2} \quad \text{(2)} \]

5. **Solve equation (1)**:
From \( \frac{-2 + a}{2} = 1 \):
\[ -2 + a = 2 \quad \Rightarrow \quad a = 2 + 2 = 4 \]

6. **Solve equation (2)**:
From \( 1 = \frac{b}{2} \):
\[ b = 2 \cdot 1 = 2 \]

7. **Final values**:
The values of \( a \) and \( b \) are:
\[ a = 4, \quad b = 2 \]

### Summary:
The values of \( a \) and \( b \) that make the points \( A(-2,-1) \), \( B(1,0) \), \( C(a,3) \), and \( D(1,b) \) form a parallelogram are:
\[ \boxed{a = 4, b = 2} \]