Given that p(1,2,-4) , Q (5,4, -6) and R (0,8,-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 break down the solution step by step.

### Step 1: Identify the coordinates of points P, Q, and R
- Point P = (1, 2, -4)
- Point Q = (5, 4, -6)
- Point R = (0, 8, -10)

### Step 2: Assume the ratio in which Q divides PR
Let the ratio in which Q divides PR be k:1, where k is the ratio of the segments on the side of P and R respectively.

### Step 3: Use the section formula
According to the section formula, the coordinates of point Q can be expressed as:
\[ Q = \left( \frac{k \cdot x_R + 1 \cdot x_P}{k + 1}, \frac{k \cdot y_R + 1 \cdot y_P}{k + 1}, \frac{k \cdot z_R + 1 \cdot z_P}{k + 1} \right) \]
Where:
- \( (x_P, y_P, z_P) = (1, 2, -4) \)
- \( (x_R, y_R, z_R) = (0, 8, -10) \)

### Step 4: Substitute the coordinates into the formula
Substituting the coordinates of points P and R into the section formula gives us:
\[ Q = \left( \frac{k \cdot 0 + 1 \cdot 1}{k + 1}, \frac{k \cdot 8 + 1 \cdot 2}{k + 1}, \frac{k \cdot (-10) + 1 \cdot (-4)}{k + 1} \right) \]
This simplifies to:
\[ Q = \left( \frac{1}{k + 1}, \frac{8k + 2}{k + 1}, \frac{-10k - 4}{k + 1} \right) \]

### Step 5: Set the coordinates of Q equal to the calculated coordinates
Since we know the coordinates of Q are (5, 4, -6), we can set up the following equations:
1. \( \frac{1}{k + 1} = 5 \)
2. \( \frac{8k + 2}{k + 1} = 4 \)
3. \( \frac{-10k - 4}{k + 1} = -6 \)

### Step 6: Solve the first equation
From the first equation:
\[ 1 = 5(k + 1) \implies 1 = 5k + 5 \implies 5k = -4 \implies k = -\frac{4}{5} \]

### Step 7: Determine the ratio
The ratio in which Q divides PR is:
\[ k:1 = -\frac{4}{5}:1 = -4:5 \]
Since we are interested in the positive ratio, we can express it as:
\[ 4:5 \]

### Final Answer
The ratio in which Q divides PR is \( 4:5 \).

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