If the points (-1, 3, 2), (-4, 2, -2) and (5, 5, y) are
collinear, then y equals

Options

A. `-10`
B. 5
C. `-5`
D. 10

Answer

To determine the value of \( y \) for which the points \((-1, 3, 2)\), \((-4, 2, -2)\), and \((5, 5, y)\) are collinear, we will follow these steps:

### Step 1: Identify the Points
Let:
- Point A: \( A(-1, 3, 2) \)
- Point B: \( B(-4, 2, -2) \)
- Point C: \( C(5, 5, y) \)

### Step 2: Find the Direction Ratios of AB and AC
The direction ratios of a line segment can be calculated using the formula:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]

**For AB:**
\[ AB = B - A = (-4 - (-1), 2 - 3, -2 - 2) = (-4 + 1, 2 - 3, -2 - 2) = (-3, -1, -4) \]

**For AC:**
\[ AC = C - A = (5 - (-1), 5 - 3, y - 2) = (5 + 1, 5 - 3, y - 2) = (6, 2, y - 2) \]

### Step 3: Set Up the Proportionality Condition
Since the points are collinear, the direction ratios of AB and AC must be proportional. Thus, we can write:
\[ \frac{-3}{6} = \frac{-1}{2} = \frac{-4}{y - 2} \]

### Step 4: Solve the Proportionality Equations
From the first two ratios:
\[ \frac{-3}{6} = \frac{-1}{2} \]
This simplifies to:
\[ -1 \cdot 6 = -3 \cdot 2 \quad \text{(which is true)} \]

Now, we will use the last ratio:
\[ \frac{-3}{6} = \frac{-4}{y - 2} \]
Cross-multiplying gives:
\[ -3(y - 2) = -4 \cdot 6 \]
\[ -3y + 6 = -24 \]
\[ -3y = -24 - 6 \]
\[ -3y = -30 \]
\[ y = \frac{-30}{-3} = 10 \]

### Conclusion
The value of \( y \) for which the points are collinear is \( y = 10 \).