If `P(x,y,z)` is a point on the line segment joining Q(2,2,4) and R(3,5,6) such that the projection 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. `1:3`

Explanation

To solve the problem, we need to find the ratio in which point P divides the line segment joining points Q(2, 2, 4) and R(3, 5, 6). We are given that the projections of the vector OP on the coordinate axes are \( \frac{13}{5} \), \( \frac{19}{5} \), and \( \frac{26}{5} \) respectively.

### Step-by-step Solution:

1. **Identify the coordinates of point P:**
The coordinates of point P are given by the projections on the axes:
\[ P\left(\frac{13}{5}, \frac{19}{5}, \frac{26}{5}\right) \]

2. **Use the section formula:**
Let P divide the line segment QR in the ratio \( \lambda : 1 \). The coordinates of point P can be expressed using the section formula:
\[ P = \left(\frac{3\lambda + 2}{\lambda + 1}, \frac{5\lambda + 2}{\lambda + 1}, \frac{6\lambda + 4}{\lambda + 1}\right) \]

3. **Set up equations for x, y, and z coordinates:**
We can set up the following equations based on the coordinates of P:
- For x-coordinate:
\[ \frac{3\lambda + 2}{\lambda + 1} = \frac{13}{5} \]
- For y-coordinate:
\[ \frac{5\lambda + 2}{\lambda + 1} = \frac{19}{5} \]
- For z-coordinate:
\[ \frac{6\lambda + 4}{\lambda + 1} = \frac{26}{5} \]

4. **Solve the first equation for λ:**
Start with the x-coordinate equation:
\[ 5(3\lambda + 2) = 13(\lambda + 1) \]
Simplifying:
\[ 15\lambda + 10 = 13\lambda + 13 \]
\[ 2\lambda = 3 \implies \lambda = \frac{3}{2} \]

5. **Verify with other coordinates:**
We can verify the value of λ with the y and z equations:
- For y-coordinate:
\[ 5\left(\frac{3}{2}\right) + 2 = \frac{19}{5}\left(\frac{3}{2} + 1\right) \]
Simplifying gives the same λ.
- For z-coordinate:
\[ 6\left(\frac{3}{2}\right) + 4 = \frac{26}{5}\left(\frac{3}{2} + 1\right) \]
Again, this confirms the value of λ.

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**.