Question

Given planes are $\begin{array}{ccc}{P}_1:cy+bz=x& P_2:az+cx=y& P_3:bx+ay=z\end{array}$ $P_1,P_2$ and $P_3$ pass through one line, if

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

Answer 1

Answer: (B)

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

Equation of the planes can be rewritten as follows:
$-x+cy+bz=0$
$cx-y+az=0$
$bx+ay-z=0$
If the given three plane intersect in a line, then
$\Delta =\left|\begin{array}{ccc}-1& c& b\\ c& -1& a\\ b& a& -1\end{array}\right|=0\Rightarrow a^2+b^2+c^2+2abc=1$