Let P(1,5,35),Q(7,5,5),R(1,λ,7),S(2λ,1,2) Given P,Q,R,S are coplanar. Then, PQ, PR, PS lie on the same plane. PQ=(7−1)
∧
i
+(5−5)
∧
j
+(5−35)
∧
k
=6
∧
i
−30
∧
k
PR=(1−1)
∧
i
+(λ−5)
∧
j
+(7−35)
∧
k
=(λ−5)
∧
j
−28
∧
k
PS=(2λ−1)
∧
i
+(1−5)
∧
j
+(2−35)
∧
k
=(2λ−1)
∧
i
−4
∧
j
−33
∧
k
∵ PQ,PR and PS lie on same plane, then |
6
0
−30
0
λ−5
−28
2λ−1
−4
−33
|=0 Expand along first row, 6[−33(λ−5)−112]+30[(2λ−1)(λ−5)]=0 ⇒6(−33λ+53)+30(2λ2−11λ+5)=0 ⇒60λ2−528λ+468=0 ⇒10λ2−88λ+78=0 ⇒5λ2−44λ+39=0 . . . (i) Possible value of λ are roots of Eq. (i). Then, sum of all possible values of λ= Sum of roots of Eq. (i) =