If a point R ( 4 , y , z) lies on the line segment
Question:

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

  1. $2 \sqrt{14}$

  2. 6

  3. $\sqrt{53}$

  4. $2 \sqrt{21}$


Correct Option: 1

Solution:

$\frac{4}{2}=\frac{-y}{y+3}=\frac{10-z}{z-4}$

$\Rightarrow \mathrm{z}=6 \& \mathrm{y}=-2$

$\Rightarrow \mathrm{R}(4,-2,6)$

dist. from origin $=\sqrt{16+4+36}=2 \sqrt{14}$

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