Let `P(2,-1,4)` and `Q(4,3,2)` are two points and as point `R` on `PQ` is such that `3PQ=5QR`, then the coordinates of `R` are

Options

A. `(14/5,3/5,16/5)`
B. `(16/5,7/5,14/5)`
C. `(11/4,1/2,13/4)`
D. none of these

Explanation

To find the coordinates of point \( R \) on line segment \( PQ \) such that \( 3PQ = 5QR \), we can follow these steps:

### Step 1: Understand the relationship between the segments
We know that \( R \) divides the line segment \( PQ \) in the ratio \( 2:3 \). This means that \( PR: RQ = 2:3 \).

### Step 2: Identify the coordinates of points \( P \) and \( Q \)
Given:
- \( P(2, -1, 4) \)
- \( Q(4, 3, 2) \)

### Step 3: Apply the section formula
The section formula for a point \( R \) dividing the line segment \( PQ \) in the ratio \( m:n \) is given by:
\[ R\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]
Where:
- \( (x_1, y_1, z_1) \) are the coordinates of point \( P \)
- \( (x_2, y_2, z_2) \) are the coordinates of point \( Q \)
- \( m = 2 \) and \( n = 3 \)

### Step 4: Substitute the values into the formula
Using the coordinates of \( P \) and \( Q \):
- \( x_1 = 2, y_1 = -1, z_1 = 4 \)
- \( x_2 = 4, y_2 = 3, z_2 = 2 \)

Now, substituting into the section formula:
\[ R_x = \frac{2 \cdot 4 + 3 \cdot 2}{2 + 3} = \frac{8 + 6}{5} = \frac{14}{5} \]
\[ R_y = \frac{2 \cdot 3 + 3 \cdot (-1)}{2 + 3} = \frac{6 - 3}{5} = \frac{3}{5} \]
\[ R_z = \frac{2 \cdot 2 + 3 \cdot 4}{2 + 3} = \frac{4 + 12}{5} = \frac{16}{5} \]

### Step 5: Write the coordinates of point \( R \)
Thus, the coordinates of point \( R \) are:
\[ R\left( \frac{14}{5}, \frac{3}{5}, \frac{16}{5} \right) \]

### Final Answer
The coordinates of point \( R \) are \( \left( \frac{14}{5}, \frac{3}{5}, \frac{16}{5} \right) \).

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