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:

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

Options

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

Explanation

To solve the problem, we need to find the coordinates of the point \( R(4, y, z) \) that lies on the line segment joining the points \( P(2, -3, 4) \) and \( Q(8, 0, 10) \), and then calculate the distance of point \( R \) from the origin.

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

The direction ratios can be found by subtracting the coordinates of point \( P \) from point \( Q \):

\[
\text{Direction ratios} = Q - P = (8 - 2, 0 - (-3), 10 - 4) = (6, 3, 6)
...