For what value of 'm' are the points A (m + 1, 1), B (2 m + 1, 3) and C (2 m + 2, 2m) collinear ?

Options

A. ` - (1)/( 2) 3`
B. ` - (1)/(2)1`
C. ` 1 (1)/(2) 2`
D. ` - (1)/(2), 2 `

Answer

To find the value of 'm' for which the points A(m + 1, 1), B(2m + 1, 3), and C(2m + 2, 2m) are collinear, we can use the concept of the area of a triangle formed by three points in a coordinate plane. If the area of the triangle is zero, then the points are collinear.

### Step-by-Step Solution:

1. **Set Up the Area Formula**:
The area of a triangle formed by three points (x1, y1), (x2, y2), and (x3, y3) can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
For our points A(m + 1, 1), B(2m + 1, 3), and C(2m + 2, 2m), we have:
- \(x_1 = m + 1\), \(y_1 = 1\)
- \(x_2 = 2m + 1\), \(y_2 = 3\)
- \(x_3 = 2m + 2\), \(y_3 = 2m\)

2. **Substitute the Points into the Area Formula**:
We substitute the coordinates into the area formula and set it to zero:
\[ \frac{1}{2} \left| (m + 1)(3 - 2m) + (2m + 1)(2m - 1) + (2m + 2)(1 - 3) \right| = 0 \]

3. **Simplify the Expression**:
We can ignore the \(\frac{1}{2}\) and the absolute value since we are setting the area to zero:
\[ (m + 1)(3 - 2m) + (2m + 1)(2m - 1) + (2m + 2)(-2) = 0 \]

4. **Expand Each Term**:
- First term:
\[ (m + 1)(3 - 2m) = 3m + 3 - 2m^2 - 2m = -2m^2 + m + 3 \]
- Second term:
\[ (2m + 1)(2m - 1) = 4m^2 - 2m + 2m - 1 = 4m^2 - 1 \]
- Third term:
\[ (2m + 2)(-2) = -4m - 4 \]

5. **Combine All Terms**:
Now we combine all the terms:
\[ -2m^2 + m + 3 + 4m^2 - 1 - 4m - 4 = 0 \]
This simplifies to:
\[ 2m^2 - 3m - 2 = 0 \]

6. **Factor the Quadratic Equation**:
To solve \(2m^2 - 3m - 2 = 0\), we can factor it:
\[ (2m + 1)(m - 2) = 0 \]

7. **Find the Values of 'm'**:
Setting each factor to zero gives us:
\[ 2m + 1 = 0 \quad \Rightarrow \quad m = -\frac{1}{2} \]
\[ m - 2 = 0 \quad \Rightarrow \quad m = 2 \]

Thus, the values of 'm' for which the points A, B, and C are collinear are:
\[ m = -\frac{1}{2} \quad \text{and} \quad m = 2 \]