If x=c y+b z, y=a z+c x, z=b x+a y where x, y, z are not all zeros, then the value of a2+b2+c2+2 a b c is

Question

If x=c y+b z, y=a z+c x, z=b x+a y where x, y, z are not all zeros, then the value of a2+b2+c2+2 a b c is (A) 0 (B) 1 (C) -1 (D) None of these

Tardigrade Admin · Accepted Answer

Given system x-c y-b z=0, c x-y+a z=0, b x+a y-z=0 System has non-trivial solution then |1&-c&-b c&-1&a b&a&-1|=0 ⇒ 1-a b c-a b c-b2-a2-c2=0 ⇒ a2+b2+c2+2 a b c=1

Q. If $x = cy + b z, y = a z + c x, z = b x + a y$ where $x, y, z$ are not all zeros, then the value of $a^{2} + b^{2} + c^{2} + 2 ab c$ is

0

1

-1

None of these

Given system $x - cy - b z = 0, c x - y + a z = 0, b x + a y - z = 0$ System has non-trivial solution then
$∣ ∣ 1 c b - c - 1 a - b a - 1 ∣ ∣ = 0$
$\Rightarrow 1 - ab c - ab c - b^{2} - a^{2} - c^{2} = 0$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 ab c = 1$

Q. If x=cy+bz,y=az+cx,z=bx+ay where x,y,z are not all zeros, then the value of a2+b2+c2+2abc is

0

1

-1

None of these

Given system x−cy−bz=0,cx−y+az=0,bx+ay−z=0 System has non-trivial solution then ∣∣​1cb​−c−1a​−ba−1​∣∣​=0 ⇒1−abc−abc−b2−a2−c2=0 ⇒a2+b2+c2+2abc=1

Q. If $x = cy + b z, y = a z + c x, z = b x + a y$ where $x, y, z$ are not all zeros, then the value of $a^{2} + b^{2} + c^{2} + 2 ab c$ is

Given system $x - cy - b z = 0, c x - y + a z = 0, b x + a y - z = 0$ System has non-trivial solution then
$∣ ∣ 1 c b - c - 1 a - b a - 1 ∣ ∣ = 0$
$\Rightarrow 1 - ab c - ab c - b^{2} - a^{2} - c^{2} = 0$
$\Rightarrow a^{2} + b^{2} + c^{2} + 2 ab c = 1$