If the system of linear equations x+2ay+az=0,   x+3by+bz=0, x+4cy+cz=0, has non-zero solution. Thena,b,c

The condition for a homogeneous system of equations has non-zero solution is Δ=0⇒12aa13bb14cc=0C2→C2-2C3⇒ 10a1bb12cc=0⇒R2→R2−R1R3→R3−R2, 10a0bb-a02c-bc-b=0 ⇒bc-b-2c-bb-a=0 ⇒bc-b2-2bc+2ca+b2-ab=0 ⇒2ca=bc+ab ⇒2b=1a+1c