Question

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. $\sqrt{53}$
• B. $2\sqrt{14}$
• C. $2\sqrt{21}$
• D. $6$

Answer 1

The correct option is B $2\sqrt{14}$

Equation of line segment joining the points $P(2,-3,4)$ and $Q(8,0,10)$ is
$\frac{x-2}{2}=\frac{y+3}{1}=\frac{z-4}{2}$
Any point on the line $PQ$ is of the form $(2\lambda +2,\lambda -3,2\lambda +4)$
Since, $R(4,y,z)$ lies on the above line segment,
$\therefore 2\lambda +2=4\Rightarrow \lambda =1$
Thus, coordinates of $R$ is $(4,-2,6)$

Distance of $R$ from the origin $=\sqrt{(4{)}^2+(-2{)}^2+(6{)}^2}=\sqrt{56}=2\sqrt{14}$