Question

The planes $x = cy + bz,y = az + cx,z = bx + ay$pass through one line, if

• A. $a + b + c = 0$
• B. $a + b + c = 1$
• C. $a^{2} + b^{2} + c^{2} = 1$
• D. $a^{2} + b^{2} + c^{2} + 2abc = 1$

Answer 1

Answer: (D)

$a^{2} + b^{2} + c^{2} + 2abc = 1$

Answer 2

The planes are concurrent, therefore

$\left| \begin{array} { c c c } - 1 & c & b \\ c & - 1 & a \\ b & a & - 1 \end{array} \right| = 0 \Rightarrow a ^ { 2 } + b ^ { 2 } + c ^ { 2 } + 2 a b c = 1$ .