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Question

Consider the planes
P1:cy+bz=x
P2:az+cx=y
P3:bx+ay=z.
P1,P2 and P3 pass through one line, if


Your Answer
A
a2+b2+c2=ab+bc+ca
Correct Answer
B
a2+b2+c2+2abc=1
Your Answer
C
a2+b2+c2=1
Your Answer
D
a2+b2+c2+2ab+2bc+2ca+2abc=1 

Solution

The correct option is B a2+b2+c2+2abc=1
Given planes
P1:cy+bzx=0
P2:az+cxy=0
P3:bx+ayz=0

Above planes can be written as 
P1:x+cy+bz=0     ...(1)
P2:cxy+az=0     ...(2)
P3:bx+ayz=0     ...(3)
This is a homogeneous system of equations.
As all planes pass through one line, it means this system has infinitely many solutions.
So, ∣ ∣1cbc1aba1∣ ∣=0

1(1a2)c(cab)+b(ac+b)=0
1+a2+c2+abc+abc+b2=0
a2+b2+c2+2abc=1

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