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. 3:1 externally
C. 2:1 internally
D. 2:1 externally

Answer

To determine the ratio in which point R divides the line segment PQ, we can use the section formula in three-dimensional geometry. The section formula states that if a point R divides the line segment joining points P and Q in the ratio \( m:n \), then the coordinates of R can be expressed as:

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

Where \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) are the coordinates of points P and Q respectively.

### Step 1: Identify the coordinates of points P, Q, and R

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

### Step 2: Set up the section formula

Let \( R \) divide \( PQ \) in the ratio \( \lambda:1 \). According to the section formula, we can express the coordinates of R as:

\[ R = \left( \frac{5\lambda + 3}{\lambda + 1}, \frac{4\lambda + 2}{\lambda + 1}, \frac{-6\lambda - 4}{\lambda + 1} \right) \]

### Step 3: Equate the coordinates of R

Now we will equate the coordinates of R with the coordinates we have:

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

2. For the y-coordinate:
\[ 8 = \frac{4\lambda + 2}{\lambda + 1} \]

3. For the z-coordinate:
\[ -10 = \frac{-6\lambda - 4}{\lambda + 1} \]

### Step 4: Solve one of the equations

Let's solve the x-coordinate equation:

\[ 9(\lambda + 1) = 5\lambda + 3 \]
Expanding gives:
\[ 9\lambda + 9 = 5\lambda + 3 \]
Rearranging gives:
\[ 9\lambda - 5\lambda = 3 - 9 \]
\[ 4\lambda = -6 \]
\[ \lambda = -\frac{3}{2} \]

### Step 5: Determine the ratio

Since \( \lambda = -\frac{3}{2} \), this means that R divides PQ externally. The ratio of PR to QR can be found as follows:

\[ PR:QR = \left| \lambda \right| : 1 = \frac{3}{2} : 1 = 3 : 2 \]

Thus, R divides PQ in the ratio \( 3:2 \) externally.

### Final Answer

R divides PQ in the ratio \( 3:2 \) externally.