If a point ` R ( 4, y, z ) ` lies on the line segment joining the points ` P ( 2, - 3, 4 ) ` and ` Q ( 8, 0, 10) ` then the distance of R from the origin is:

Options

A. 6
B. ` sqrt ( 53 ) `
C. `2 sqrt (14 ) `
D. ` 2 sqrt ( 21 ) `

Explanation

To find the distance of the point \( R(4, y, z) \) from the origin, we first need to determine the coordinates \( y \) and \( z \) of point \( R \) based on its position on the line segment joining points \( P(2, -3, 4) \) and \( Q(8, 0, 10) \).

### Step 1: Find the direction ratios of the line segment \( PQ \)

The direction ratios can be calculated as follows:
- For \( P(2, -3, 4) \) and \( Q(8, 0, 10) \):
\[
\text{Direction ratios} = (8 - 2, 0 - (-3), 10 - 4) = (6, 3, 6)
...