Given that, a ^2+2 b +c 0 and that the system of equations aligned & (a +b) x+a y+b z=0 & (b +c) x+b y+c z=0 & (a +b) y+(b +c) z=0 aligned has a non-trivial solution, then a, b and c lie in

Step-by-step solution

Given system of equations is aligned (a +b) x+a y+b z & =0 (b +c) x+b y+c z & =0 aligned and (a +b) y+(b +c) z=0 For non-trivial solution, | array ccc a +b & a & b b +c & b & c 0 & a +b & b +c array |=0 Applying R₃ R₃- R₁-R₂ aligned & | array ccc a +b & a & b b +c & b & c - (a ^2+2 b +c ) & 0 & 0 array |=0 & - (a ^2+2 b +c ) (a c-b^2 )=0 aligned array ll a c-b^2=0 & ( a ^2+2 b +c 0 ) a c=b^2 array Hence, a, b and c are in GP.