If the points (5, 2, 4), (6, -1, 2) and (8, -7, k) are collinear, then k =

Options

A. `-1`
B. 3
C. 2
D. `-2`

Answer

To determine the value of \( k \) such that the points \( A(5, 2, 4) \), \( B(6, -1, 2) \), and \( C(8, -7, k) \) are collinear, we can follow these steps:

### Step 1: Find the direction ratios of line AB
The direction ratios of a line segment between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \) can be calculated using the formula:
\[ \text{Direction Ratios} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \]
For points \( A(5, 2, 4) \) and \( B(6, -1, 2) \):
- \( x \) direction ratio: \( 6 - 5 = 1 \)
- \( y \) direction ratio: \( -1 - 2 = -3 \)
- \( z \) direction ratio: \( 2 - 4 = -2 \)

Thus, the direction ratios of line \( AB \) are \( (1, -3, -2) \).

### Step 2: Find the direction ratios of line BC
Now, we calculate the direction ratios for line segment \( BC \) between points \( B(6, -1, 2) \) and \( C(8, -7, k) \):
- \( x \) direction ratio: \( 8 - 6 = 2 \)
- \( y \) direction ratio: \( -7 - (-1) = -6 \)
- \( z \) direction ratio: \( k - 2 \)

Thus, the direction ratios of line \( BC \) are \( (2, -6, k - 2) \).

### Step 3: Set up the condition for collinearity
For points \( A, B, C \) to be collinear, the direction ratios must be proportional. Therefore, we can set up the following ratios:
\[ \frac{1}{2} = \frac{-3}{-6} = \frac{-2}{k - 2} \]

### Step 4: Simplify the ratios
From the first two ratios:
\[ \frac{1}{2} = \frac{1}{2} \quad \text{(which is true)} \]

Now, we will focus on the last ratio:
\[ \frac{-2}{k - 2} = \frac{1}{2} \]

### Step 5: Cross-multiply to solve for \( k \)
Cross-multiplying gives:
\[ -2 \cdot 2 = 1 \cdot (k - 2) \]
\[ -4 = k - 2 \]

### Step 6: Solve for \( k \)
Adding 2 to both sides:
\[ k = -4 + 2 \]
\[ k = -2 \]

Thus, the value of \( k \) is \( -2 \).

### Final Answer:
\[ k = -2 \]

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