From a point `P(lambda,lambda,lambda)`, perpendicular PQ and PR are drawn respectively on the lines `y=x, z= 1` and `y=-x, z=-1`.If P is such that `angleQPR` is a right angle, then the possible value(s) of `lambda` is/(are)

Options

A. `sqrt(2)`
B. 1
C. -1
D. `-sqrt(2)`

Answer

C

Explanation

(i) Direction ratios of a line joining two points `(x_(1), y_(1), z_(1))` and `(x_(2), y_(2), z_(2))` are `x_(2)-x_(1), y_(2)-y_(1), z_(2)-z_(1).`
(ii) If the two lines with direction ratios `a_(1),b_(1)c_(1),a_(2),b_(2),c_(2)` are perpendicular, then` a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2)=0`
Line `L_(1)` is given by y=x,z=1 can be expressed
`L_(1):(x)/(1)=(y)/(1)=(z-1)/(0)=alpha" "["say"]`
`implies" "x=alpha,y=alpha, z=1`
Let the corrdinates of Q on `L_(1)` be `(alpha, alpha,1)`.
Line `L_(2)` given by `y=-x,z=-1` can be expressed as
`L_(2):(x)/(1)=(y)/(-1)=(z+1)/(0)=beta" "["say"]`
`implies" "x=beta,y=-beta,z=-1`
Let the coordinates of R on `L_(2)` be `(beta,-beta,-1)`.
Direction ratios of PQ are `lambda-alpha, lambda-alpha, lambda-1.`
Now, `PQ_|_L_(1)`

`:." "1(lambda-alpha)+1.(lambda-alpha)+0.(lambda-1)=0implieslambda=alpha`
Hence, Q`(lambda, lambda, 1)`
Direction retios of PR are `lambda-beta,lambda+beta,lambda+1.`
Now, `PR_|_L_(2)`
`:." "1(lambda-beta)+(-1)(lambda+beta)+0(lambda+1)=0`
`lambda-beta-lambda=0`
`implies" "beta=0`
Hence, `R(0, 0, -1)`
Now, as `angleQPR=90^(@)`
[as `a_(1)a_(2)+b_(1)b_(2)+c_(1)c_(2)=0`, if two lines with DR's `a_(1),b_(1),c_(1),a_(2),b_(2),c_(2)` are perpendicular]
`:." "(lambda-lambda)(lambda-0)+(lambda-lambda)(lambda-0)+(lambda-1)(lambda+1)=0`
`implies" "(lambda-1)(lambda+1)=0`
`implies" "lambda=1`
or`" "lambda=-1`
`lambda=1`, rejected as P and Q are different points.
`implies" "lambda=-1`