Given the points A (2, 3), B (-5, O) and C (-2, a) are collinear. Find 'a'.

Answer

To find the value of 'a' such that the points A(2, 3), B(-5, 0), and C(-2, a) are collinear, we can follow these steps:

### Step 1: Understand the concept of collinearity
Three points are collinear if the slope between any two pairs of points is the same.

### Step 2: Calculate the slope of line segment AB
The slope (m) between two points (x1, y1) and (x2, y2) is given by the formula:
\[ m = \frac{y2 - y1}{x2 - x1} \]

For points A(2, 3) and B(-5, 0):
- \( x1 = 2, y1 = 3 \)
- \( x2 = -5, y2 = 0 \)

Substituting these values into the slope formula:
\[ m_{AB} = \frac{0 - 3}{-5 - 2} = \frac{-3}{-7} = \frac{3}{7} \]

### Step 3: Calculate the slope of line segment BC
For points B(-5, 0) and C(-2, a):
- \( x1 = -5, y1 = 0 \)
- \( x2 = -2, y2 = a \)

Substituting these values into the slope formula:
\[ m_{BC} = \frac{a - 0}{-2 - (-5)} = \frac{a}{-2 + 5} = \frac{a}{3} \]

### Step 4: Set the slopes equal to each other
Since points A, B, and C are collinear, we set the slopes equal:
\[ m_{AB} = m_{BC} \]
\[ \frac{3}{7} = \frac{a}{3} \]

### Step 5: Cross-multiply to solve for 'a'
Cross-multiplying gives:
\[ 3 \cdot 3 = 7 \cdot a \]
\[ 9 = 7a \]

### Step 6: Solve for 'a'
To find 'a', divide both sides by 7:
\[ a = \frac{9}{7} \]

### Final Answer
The value of 'a' is \( \frac{9}{7} \).

---