If the system of linear equations $x+2 a y+a z=0$, $x+3 b y+b z=0, x+4 c y+c z=0$ has a non-zero sol

Question

If the system of linear equations $x+2 a y+a z=0$, $x+3 b y+b z=0, x+4 c y+c z=0$ has a non-zero solution, then $a, b, c$\

Professor PC Thomas Classes · Answer

(b) : For non trivial solution the determinant of the coefficient of various term vanish\ i.e. $\left|\begin{array}{lll}1 & 2 a & a \ 1 & 3 b & b \ 1 & 4 c & c\end{array}ight|=0$\ $\Rightarrow(3 b c-4 b c)-2 a(c-b)+a(4 c-3 b)=0$\ $\Rightarrow \frac{2 a c}{a+c}=b \Rightarrow a, b, c \in$ H.P.\