If `P(3,2,-4),Q(5,4,-6)` and `R(9,8,-10)` are collinear, then `R` divides `PQ` in the ratio

Options

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

Explanation

To determine the ratio in which point \( R(9, 8, -10) \) divides the line segment \( PQ \) where \( P(3, 2, -4) \) and \( Q(5, 4, -6) \) are given, we can use the section formula in three-dimensional geometry.

The section formula states that if a point \( R(x, y, z) \) divides the line segment joining points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) in the ratio \( m:n \), then the coordinates of \( R \) can be expressed as:

\[ R = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

### Step 1: Set up the equations for the coordinates of \( R \)

Given:
- \( P(3, 2, -4) \)
- \( Q(5, 4, -6) \)
- \( R(9, 8, -10) \)

Let \( R \) divide \( PQ \) in the ratio \( m:n \). We can set up the following equations based on the coordinates:

1. For the x-coordinate:
\[ 9 = \frac{m \cdot 5 + n \cdot 3}{m+n} \]

2. For the y-coordinate:
\[ 8 = \frac{m \cdot 4 + n \cdot 2}{m+n} \]

3. For the z-coordinate:
\[ -10 = \frac{m \cdot (-6) + n \cdot (-4)}{m+n} \]

### Step 2: Solve the equations

#### For the x-coordinate:
\[ 9(m+n) = 5m + 3n \]
\[ 9m + 9n = 5m + 3n \]
\[ 4m + 6n = 0 \quad \text{(Equation 1)} \]

#### For the y-coordinate:
\[ 8(m+n) = 4m + 2n \]
\[ 8m + 8n = 4m + 2n \]
\[ 4m + 6n = 0 \quad \text{(Equation 2)} \]

#### For the z-coordinate:
\[ -10(m+n) = -6m - 4n \]
\[ -10m - 10n = -6m - 4n \]
\[ -4m - 6n = 0 \quad \text{(Equation 3)} \]

### Step 3: Analyze the equations

From Equation 1 and Equation 2, we have:
\[ 4m + 6n = 0 \]
This can be simplified to:
\[ 2m + 3n = 0 \quad \Rightarrow \quad \frac{m}{n} = -\frac{3}{2} \]

### Step 4: Conclusion

Thus, the ratio \( m:n \) is \( 3:2 \). Therefore, point \( R \) divides the line segment \( PQ \) in the ratio \( 3:2 \).