Find the value of `lambda` so that the points `P ,Q ,Ra n dS` on the sides `O A ,O B ,O Ca n dA B ,` respectively, of a regular tetrahedron `O A B C` are coplanar. It is given that `(O P)/(O A)=1/3,(O Q)/(O B)=1/2,(O R)/(O C)=1/3a n d(O S)/(A B)=lambdadot` a. `lambda=1/2` b. `lambda=-1` c. `lambda=0` d. for no value of`lambda`

Options

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

Answer

B

Explanation

Let `vec(OA) = veca, vec(OB) = vecb and vec(OC) = vecc`,
then `vec(AB) = vecb - veca and vec(OP) =(1)/(3) veca`,
`" "vec(OQ) = (1)/(2) vecb, vec(OR) = (1)/(3) vecc`.
Since P, Q, R and S are coplanar, then
`vec(PS) = alpha vec(PQ) + betavec(PR) (vec(PS) ` can be written as a linear combination of `vec(PQ) and vec(PR))`
`= alpha (vec(OQ) - vec(OP))+ beta(vec(OR) - vec(OP))`
i.e., `vec(OS) - vec(OP) = -(alpha + beta)( veca)/(3) + (alpha )/(2) vecb + (beta)/(3) vecc`
`rArr vec(OS) = (1-alpha- beta) (veca)/(3) + (alpha)/(2) vecb+ (beta)/(3) vecc" "` (i)
Given `vec(OS) =lamda (vecb -veca) " "` (ii)
From (i) and (ii), `beta =0, (1-alpha)/(3) = -lamda and (alpha)/(2) =lamda`
`rArr 2 lamda = 1+ 3lamda`
or `lamda =-1 `