# If a point $R\left ( 4,y,z \right )$ lies on the line segment joining the points $P\left ( 2,-3,4 \right )$ and $Q\left ( 8,0,10 \right ),$ then distance of R from the origin is : Option 1)$2\sqrt{14}$Option 2)$2\sqrt{21}$Option 3)$6$  Option 4)$\sqrt{53}$

$P(2,-3,4)$

$Q(8,0,10)$

$R(4,y,z)$

Direction Ratio of PQ= $(6,3,6)$

Direction ratio of PR= $(2,y+3,z-4)$

P,R and Q lie on same line

$\frac{6}{2}=\frac{3}{y+3}=\frac{6}{z-4}$

$y=-2\: \: ,\: z=6$

Point $R=\left ( 4,-2,6 \right )$

Distance from origin

$\sqrt{16+4+36}=\sqrt{56}$

$=2\sqrt{14}$

Option 1)

$2\sqrt{14}$

Option 2)

$2\sqrt{21}$

Option 3)

$6$

Option 4)

$\sqrt{53}$

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