Question

If x = cy + bz, y = az + cx, z = bx + ay (where x, y, z are not all zero) have a solution other than x = 0, y = 0, z = 0, then a, b and c are connected by the relation

• A. $a^2+b^2+c^2+3abc=0$
• B. $a^2+b^2+c^2+2abc=0$
• C. $a^2+b^2+c^2+2abc=1$
• D. $a^2+b^2+c^2-bc-ca-ab=1$

Answer 1

The correct option is C $a^2+b^2+c^2+2abc=1$

The system of homogeneous equations
x - cy - bz = 0
cx - y + az = 0
bx + ay - z = 0
has a non-trivial solution (since x, y, z are not all zero)
If $\Delta =\left|\begin{array}{ccc}1& -c& -b\\ c& -1& a\\ b& a& -1\end{array}\right|=0$
i.e., if $(1-a^2)+c(-c-ab)-b(ac+b)=0$
i.e., if $a^2+b^2+c^2+2abc=1$