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 coordinates of point \( P(x,y,z) \) and determine the ratio in which it divides the line segment joining points \( Q(2,2,4) \) and \( R(3,5,6) \).

### Step 1: Identify the coordinates of point \( P \)

Given the projections of \( \vec{OP} \) on the axes are:
- \( x \)-coordinate: \( \frac{13}{5} \)
- \( y \)-coordinate: \( \frac{19}{5} \)
- \( z \)-coordinate: \( \frac{26}{5} \)

Thus, the coordinates of point \( P \) are:
\[ P\left(\frac{13}{5}, \frac{19}{5}, \frac{26}{5}\right) \]

### Step 2: Set up the section formula

Let \( P \) divide \( QR \) in the ratio \( k:1 \). According to the section formula, the coordinates of point \( P \) can be expressed as:
\[ P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]
where \( Q(x_1, y_1, z_1) = (2, 2, 4) \) and \( R(x_2, y_2, z_2) = (3, 5, 6) \).

### Step 3: Substitute the coordinates into the section formula

Using the section formula, we can write:
\[ P\left(\frac{13}{5}, \frac{19}{5}, \frac{26}{5}\right) = \left( \frac{3k + 2}{k+1}, \frac{5k + 2}{k+1}, \frac{6k + 4}{k+1} \right) \]

### Step 4: Set up equations for each coordinate

From the \( x \)-coordinate:
\[ \frac{3k + 2}{k + 1} = \frac{13}{5} \]
Cross-multiplying gives:
\[ 5(3k + 2) = 13(k + 1) \]
Expanding both sides:
\[ 15k + 10 = 13k + 13 \]
Rearranging gives:
\[ 2k = 3 \implies k = \frac{3}{2} \]

### Step 5: Verify with the \( y \)-coordinate

Now, let's check the \( y \)-coordinate:
\[ \frac{5k + 2}{k + 1} = \frac{19}{5} \]
Substituting \( k = \frac{3}{2} \):
\[ \frac{5\left(\frac{3}{2}\right) + 2}{\frac{3}{2} + 1} = \frac{\frac{15}{2} + 2}{\frac{5}{2}} = \frac{\frac{15 + 4}{2}}{\frac{5}{2}} = \frac{\frac{19}{2}}{\frac{5}{2}} = \frac{19}{5} \]
This confirms that the \( y \)-coordinate is correct.

### Step 6: Verify with the \( z \)-coordinate

Now, let's check the \( z \)-coordinate:
\[ \frac{6k + 4}{k + 1} = \frac{26}{5} \]
Substituting \( k = \frac{3}{2} \):
\[ \frac{6\left(\frac{3}{2}\right) + 4}{\frac{3}{2} + 1} = \frac{9 + 4}{\frac{5}{2}} = \frac{13}{\frac{5}{2}} = \frac{26}{5} \]
This confirms that the \( z \)-coordinate is also correct.

### Conclusion

Thus, point \( P \) divides the line segment \( QR \) in the ratio:
\[ \frac{3}{2} : 1 \quad \text{or} \quad 3 : 2 \]