If $x^2+y^2+z^2
eq 0, \quad x=c y+b z, \quad y=a z+c x$ and $z=b x+a y$, then $a^2+b^2+c^2+2 a b c$ is equal to

Question

If $x^2+y^2+z^2 
eq 0, \quad x=c y+b z, \quad y=a z+c x$ and $z=b x+a y$, then $a^2+b^2+c^2+2 a b c$ is equal to

Aicharya · Accepted Answer

We have, $ \begin{aligned} x-c y-b z & =0 \ -c x+y-a z & =0 \ -b x-a y+z & =0 \end{aligned} $ Eliminating $x, y, z$, we get $ \begin{array}{rr} & \left|\begin{array}{ccc} 1 & -c & -b \ -c & 1 & -a \ -b & -a & 1 \end{array}ight|=0 \ \Rightarrow & 1\left(1-a^2ight)+c(-c-a b)-b(c a+b)=0 \ \Rightarrow & 1-a^2-c^2-a b c-a b c-b^2=0 \ \Rightarrow & a^2+b^2+c^2+2 a b c=1 \end{array} $

If $x^2+y^2+z^2 \neq 0, \quad x=c y+b z, \quad y=a z+c x$ and $z=b x+a y$, then $a^2+b^2+c^2+2 a b c$ is…

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