The value of the` lambda` so that P, Q, R, S on the sides OA, OB, OC and AB of a regular tetrahedron are coplanar. When `(OP)/(OA)=1/3 ;(OQ)/(OB)=1/2` and `(OS)/(AB)=lambda` is (A) `lamda=1/2` (B) `lamda=-1` (C) `lamda=0` (D) `lamda=2`

Options

A. `lamda=1/2`
B. `lamda=-1`
C. `lamda=0`
D. fo no value of `lamda`

Answer

B

Explanation

Let `OA=a,OB=b and OC=c`.
then `AB=b-a and OP=(1)/(3)a`.
`OQ=(1)/(2)b,OR=(1)/(3)c`.
Since, P,Q,R and S are coplanar, then
`PS=alphaPQ+betaPR`
(PS can be written as a linear combination of PQ and PR)
`=alpha(OQ-OP)+beta(OR-OP)`
i.e., `OS-OP=-(alpha+beta)(a)/(3)+(alpha)/(2)b+(beta)/(3)c`
`implies OS=(1-alpha-beta)(a)/(3)+(alpha)/(2)b+(beta)/(3)c` . . (i)
Given `OS=lamdaAB=lamda(b-a)` . . . (ii)
From Eq. (i) and Eq. (ii), `beta=0,(1-alpha)/(3)=-lamda and (alpha)/(2)=lamda`.
`implies 2 lamda=1+3lamda` or `lamda=-1`.