Find the value of 'a' for which the following points A (a, 3), B (2, 1) and C (5, a) are collinear. Hence, find the equation of the line.

Answer

To find the value of 'a' for which the points A(a, 3), B(2, 1), and C(5, a) are collinear, we can use the concept of slopes. If the points are collinear, the slope between any two pairs of points must be the same.

### Step-by-Step Solution:

1. **Identify the Points**:
- A = (a, 3)
- B = (2, 1)
- C = (5, a)

2. **Calculate the Slope of AB**:
The slope of line segment AB is given by the formula:
\[ \text{slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 3}{2 - a} = \frac{-2}{2 - a} \]

3. **Calculate the Slope of BC**:
The slope of line segment BC is given by:
\[ \text{slope of BC} = \frac{a - 1}{5 - 2} = \frac{a - 1}{3} \]

4. **Set the Slopes Equal**:
Since the points are collinear, we set the slopes equal:
\[ \frac{-2}{2 - a} = \frac{a - 1}{3} \]

5. **Cross-Multiply**:
Cross-multiplying gives:
\[ -2 \cdot 3 = (a - 1)(2 - a) \]
\[ -6 = (a - 1)(2 - a) \]

6. **Expand the Right Side**:
Expanding the right side:
\[ -6 = 2a - a^2 - 2 + a \]
\[ -6 = -a^2 + 3a - 2 \]

7. **Rearrange the Equation**:
Rearranging gives:
\[ a^2 - 3a - 4 = 0 \]

8. **Factor the Quadratic**:
Factoring the quadratic:
\[ (a - 4)(a + 1) = 0 \]

9. **Find the Values of 'a'**:
Setting each factor to zero gives:
\[ a - 4 = 0 \quad \Rightarrow \quad a = 4 \]
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

10. **Conclusion**:
The values of 'a' for which the points A, B, and C are collinear are \( a = 4 \) and \( a = -1 \).

### Find the Equation of the Line:

**For \( a = -1 \)**:
- Points: A(-1, 3), B(2, 1), C(5, -1)
- Using the two-point form of the line equation with points A and B:
\[ \frac{y - 3}{x + 1} = \frac{1 - 3}{2 - (-1)} = \frac{-2}{3} \]
Cross-multiplying gives:
\[ 3(y - 3) = -2(x + 1) \]
Simplifying:
\[ 3y - 9 = -2x - 2 \quad \Rightarrow \quad 2x + 3y = 7 \]

**For \( a = 4 \)**:
- Points: A(4, 3), B(2, 1), C(5, 4)
- Using the two-point form of the line equation with points A and C:
\[ \frac{y - 3}{x - 4} = \frac{4 - 3}{5 - 4} = 1 \]
Cross-multiplying gives:
\[ y - 3 = x - 4 \quad \Rightarrow \quad x - y = -1 \]

### Final Results:
- The values of 'a' are \( a = 4 \) and \( a = -1 \).
- The equations of the lines are:
1. For \( a = -1 \): \( 2x + 3y = 7 \)
2. For \( a = 4 \): \( x - y = -1 \)