If `P(x,y,z)` is a point on the line segment joining `Q(2,2,4)` and `R(3,5,6)` such that projections of `vec(OP)` on the axes are `13/5,19/5,26/5` respectively, then `P` divides `QR` in the ratio

Options

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

Explanation

To solve the problem, we need to find the point \( P(x,y,z) \) that divides the line segment joining points \( Q(2,2,4) \) and \( R(3,5,6) \) in a certain ratio, given the projections of \( \vec{OP} \) on the axes.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points Q and R**:
- \( Q(2, 2, 4) \)
- \( R(3, 5, 6) \)

2. **Determine the Coordinates of Point P**:
- The coordinates of point \( P \) can be expressed in terms of the ratio \( \lambda : 1 \) as follows:
\[ P(x, y, z) = \left( \frac{3\lambda + 2}{\lambda + 1}, \frac{5\lambda + 2}{\lambda + 1}, \frac{6\lambda + 4}{\lambda + 1} \right) \]

3. **Set Up the Equations Based on Projections**:
- We know the projections of \( \vec{OP} \) on the axes are given as:
\[ x = \frac{13}{5}, \quad y = \frac{19}{5}, \quad z = \frac{26}{5} \]
- This gives us three equations:
\[ \frac{3\lambda + 2}{\lambda + 1} = \frac{13}{5} \quad (1) \]
\[ \frac{5\lambda + 2}{\lambda + 1} = \frac{19}{5} \quad (2) \]
\[ \frac{6\lambda + 4}{\lambda + 1} = \frac{26}{5} \quad (3) \]

4. **Solve the First Equation**:
- From equation (1):
\[ 5(3\lambda + 2) = 13(\lambda + 1) \]
\[ 15\lambda + 10 = 13\lambda + 13 \]
\[ 2\lambda = 3 \implies \lambda = \frac{3}{2} \]

5. **Verify with the Other Equations**:
- Substitute \( \lambda = \frac{3}{2} \) into equations (2) and (3) to verify:
- For equation (2):
\[ \frac{5(\frac{3}{2}) + 2}{\frac{3}{2} + 1} = \frac{\frac{15}{2} + 2}{\frac{5}{2}} = \frac{\frac{19}{2}}{\frac{5}{2}} = \frac{19}{5} \]
- For equation (3):
\[ \frac{6(\frac{3}{2}) + 4}{\frac{3}{2} + 1} = \frac{9 + 4}{\frac{5}{2}} = \frac{13}{\frac{5}{2}} = \frac{26}{5} \]
- Both equations are satisfied.

6. **Determine the Ratio**:
- Since \( \lambda = \frac{3}{2} \), the ratio in which \( P \) divides \( QR \) is:
\[ \text{Ratio} = \lambda : 1 = \frac{3}{2} : 1 = 3 : 2 \]

### Final Answer:
The point \( P \) divides the line segment \( QR \) in the ratio \( 3 : 2 \).