If a ,   b ,   c are non - zero real numbers and if the system of equations a - 1 x = y + z b - 1…

If a, b, c  are non - zero real numbers and if the system of equations

a-1x=y+z

b-1y=x+z

c-1z=x+y

has a non - trivial solution, then ab+bc+ca  equals :

  1. -1
  2. a+b+c
  3. abc
  4. 1

Solution

Given, a-1x=y+z⇒a-1x-y-z=0

b-1y=z+x⇒-x+b-1y-z=0

c-1z=y+x=-x-y+c-1z=0

For a non-trivial solution

a-1-1-1-1b-1-1-1-1c-1=0

⇒   1-a1111-b1111-c=0

⇒   1-a1-b1-c-1-11-c-1+1-1-b=0

⇒   1-a+b+c+ab+bc+ca-abc-1+a+b+c=0

⇒ab+bc+ca=abc

Asked in: JEE Main 2014 (09 Apr Online)

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