If the system of linear equations
`ax +by+ cz =0`
`cx + cy + bz =0`
` bx + cy +az =0` where a,b,c,` in` R are non-zero and distinct, has a non-zero solution, then:

Options

A. `a +b+c=0`
B. `a,b,c` are in A.P.
C. `1/a, 1/b, 1/c` are in A.P.
D. a,b,c are in G.P.

Answer

A

Explanation

`|{:(a,b,c),(c,a,b),(b,c,a):}|=0`
`implies(a ^(2) -bc) -b (ac - b ^(2)) + c (c ^(2) - ab) =0 implies a ^(3) -abc -abc + b ^(3) + c ^(3) -abc = 0`
`implies a ^(3) + b ^(3)+ c ^(3) -3abc =0 implies (a +b+c) (a^(2) + b ^(2) + c ^(2)-ab-b-ca)=0`
`implies a+b+c=0`