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If P(x ,y ,z) is a point on the line ...

If `P(x ,y ,z)` is a point on the line segment joining `Q(2,2,4)a n d R(3,5,6)` such that the projections of ` vec O P` on te axes are 13/5, 19/5 and 26/5, respectively, then find the ratio in which `P` divides `Q Rdot`

A

`1:2`

B

`3:2`

C

`2:3`

D

`1:3`

Text Solution

Verified by Experts

The correct Answer is:
B
Since `vec(OP)` has projection `13/5, 19/5` and `26/5` on the co-ordinate axes, therefore
`vec(OP) = 13/5hati+19/5hatj+26/5hatk`
Suppose P divides the line segment joining Q(2,2,4) and Q(3,5,6) in the ratio `lambda:1`
Then the position vector of P is
`((3lambda+2)/(lambda+1))hati+((5lambda+2)/(lambda+1)hatj+(6lambda+4)/(lambda+1)hatk)`
`therefore 13/5hati+19/5hatj+26/5hatk`
`=(3lambda+2)/(lambda+1)hati+(5lambda+2)/(lambda+1)hatj+(6lambda+4)/(lambda+1)hatk`
`rArr (3lambda+2)/(lambda+1)=13/5, (5lambda+2)/(lambda+1)=19/5, (6lambda+4)/(lambda+1)=26/5`
`rArr lambda=3/2`
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