If the points `A(3, 2, -4), B(9, 8, -10) and C(-2, -3. p)` are collinear, then p=

Options

A. `9`
B. `-9`
C. `1`
D. `-1`

Answer

To determine the value of \( p \) such that the points \( A(3, 2, -4) \), \( B(9, 8, -10) \), and \( C(-2, -3, p) \) are collinear, we can follow these steps:

### Step 1: Find the direction ratios of vectors \( \overrightarrow{AB} \) and \( \overrightarrow{BC} \)

The direction ratio of vector \( \overrightarrow{AB} \) is calculated as follows:

\[ \overrightarrow{AB} = B - A = (9 - 3, 8 - 2, -10 - (-4)) = (6, 6, -6) \]

Now, we calculate the direction ratio of vector \( \overrightarrow{BC} \):

\[ \overrightarrow{BC} = C - B = (-2 - 9, -3 - 8, p - (-10)) = (-11, -11, p + 10) \]

### Step 2: Set the direction ratios equal

Since the points are collinear, the direction ratios must be proportional:

\[ \frac{6}{-11} = \frac{6}{-11} = \frac{-6}{p + 10} \]

### Step 3: Cross-multiply to solve for \( p \)

From the first two ratios, we can simplify:

\[ \frac{6}{-11} = \frac{-6}{p + 10} \]

Cross-multiplying gives:

\[ 6(p + 10) = -6(-11) \]

This simplifies to:

\[ 6(p + 10) = 66 \]

### Step 4: Solve for \( p \)

Now, divide both sides by 6:

\[ p + 10 = 11 \]

Subtracting 10 from both sides gives:

\[ p = 1 \]

### Conclusion

Thus, the value of \( p \) is:

\[ \boxed{1} \]