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Let P be the point (1, 2, 3) and Q be a ...

Let P be the point (1, 2, 3) and Q be a point on the line `hat(r )=(hat(i)-hat(j)+5hat(k))+ lambda (-2hat(i)+3hat(j)+4hat(k))`. Then the value of `lambda` for which line PQ is perpendicular to the plane `4x + 9y – 18z = 1` is :

A

`(1)/(3)`

B

`-(1)/(6)`

C

`-(2)/(3)`

D

`(2)/(5)`

Text Solution

Verified by Experts

The correct Answer is:
D
P(1, 2, 3)
`Q(1-2lambda, -1 + 3lambda, 5 + 4lambda)`
`vec(PQ)=vec(Q)-vec(P)`
`=(-2lambda)hat(i)+(-3+3lambda)hat(j)+(2+4lambda)hat(k)`
`var(PQ)` is perpendicular to plane `4x + 9y - 18z = 1`
`therefore vec(PQ) xx vec(n)|(i,j,k),(-2lambda,-3+3lambda,2+4lambda),(4,9,-10)|=0`
`54-54lambda-18-36lambda-36lambda+8+16lambda-18lambda+12-12lambda = 0`
`lambda = (2)/(5)`
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