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Question

The planes x-cy-bz=0, cx-y+az=0 and bx+ay-z=0 pass through a straight line, where a,b,c are non-zero constants. Then the value of a2+b2+c2+2abc is


A

-1

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B

2

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C

1

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D

0

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Solution

The correct option is C

1


Given planes are:

x-cy-bz=0 ...................(i)
cx-y+az=0 ...............(ii)
bx+ay-z=0 .................(iii)

Equation pf planes passing through the line of intersection of plane (i) and (ii) may be taken as;
(xcybz)+λ(cxy+az)=0i.e. x(1+λc)y(c+λ)+z(b+aλ)=0...........(iv)
If plane (iii) and (iv) are the same, then equation (iii) and (iv) will be identical.
1+xλb=(c+λ)a=b+aλ1λ=(a+bc)ac+b and λ=(ab+c)1a2(a+bc)(ac+b)=(ab+c)(1a2)aa3+bca2bc=a2bc+ac2+ab2+bc2a2bc+ac2+ab2+a3a=0a(2abc+c2+b2+a21)=0a2+b2+c2+2abc=1 ( a0)


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