Three Dimensional Geometry Part 2

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)
=‌

−(−44)

5

=‌

44

5

[∵ax2+bx+ +c=0, sum of roots =−b∕a}