Given that p(3,2,-4) , Q (5,4, -6) and R (9,8,-10) are collinear find the ratio in which Q divides PR

Options

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

Explanation

To find the ratio in which point Q divides the line segment PR, we can use the section formula. Let's go through the solution step by step.

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( P(3, 2, -4) \)
- Let \( Q(5, 4, -6) \)
- Let \( R(9, 8, -10) \)

2. **Assume the Ratio**:
- Let \( Q \) divide \( PR \) in the ratio \( k:1 \).

3. **Use the Section Formula**:
- The section formula states that if a point \( Q \) divides the line segment joining points \( P(x_1, y_1, z_1) \) and \( R(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of \( Q \) are given by:
\[ Q\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]
- Here, \( m = k \) and \( n = 1 \).

4. **Substitute the Coordinates**:
- For the x-coordinate:
\[ 5 = \frac{k \cdot 9 + 1 \cdot 3}{k + 1} \]
- For the y-coordinate:
\[ 4 = \frac{k \cdot 8 + 1 \cdot 2}{k + 1} \]
- For the z-coordinate:
\[ -6 = \frac{k \cdot (-10) + 1 \cdot (-4)}{k + 1} \]

5. **Solve for k using the x-coordinate**:
- Multiply both sides by \( (k + 1) \):
\[ 5(k + 1) = 9k + 3 \]
- Expanding gives:
\[ 5k + 5 = 9k + 3 \]
- Rearranging gives:
\[ 5 - 3 = 9k - 5k \implies 2 = 4k \implies k = \frac{1}{2} \]

6. **Determine the Ratio**:
- The ratio in which \( Q \) divides \( PR \) is \( k:1 = \frac{1}{2}:1 \).
- This can be expressed as \( 1:2 \).

### Final Answer:
The ratio in which \( Q \) divides \( PR \) is \( 1:2 \).