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A point R with x-coordinate 4 lies on th...

A point R with x-coordinate 4 lies on the line segment joining the points `P(2,-3,4)` and `Q(8,0,10)`. Find the coordinates of the point R.

A

`2sqrt(21)`

B

`sqrt(53)`

C

`2sqrt(14)`

D

`6`

Text Solution

Verified by Experts

Given points are `P(2,-3,4)`, `Q(8,0,10)` and `R(4,y,z)`. Now, equation of line passing through points `P` and `Q` is `(x-8)/(6)=(y-0)/(3)=(z-10)/(6)`
[Since equation of a line passing through two points
`A(x_(1),y_(1),z_(1))` and `B(x_(2),y_(2)z_(2))` is given by
`(x-x_(1))/(x_(2)-x_(1))=(y-y_(1))/(y_(2)-y_(1))=(z-z_(1))/(z_(2)-z_(1))]`
`implies(x-8)/(2)=(y)/(1)=(z-10)/(2)`......`(i)`
`:'` Points `P,Q` and `R` are collinear, so
`(4-8)/(2)=(y)/(1)=(z-10)/(2)`
`implies-2=y=(z-10)/(2)`
`impliesy=-2`
and `z=6`
So, point `R` is `(4,-2,-6)`, therefore the distance of point `R` from origin is
`OR=sqrt(16+4+36)`
`=sqrt(56)=2sqrt(14)`
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