Question

If the system of linear equation $x+2ay+az=0$, $x+3by+bz=0$, $x+4cy+cz=0$ has a non-zero solution, then $a,b,c$ are in

• A. $G.P.$
• B. $H.P.$
• C. $A.P$
• D. None of the above

Answer 1

The correct option is B $H.P.$

For the system of linear equation to have non-zero solution, the determinant of coefficient should be zero.

$\left|\begin{array}{ccc}1& 2a& a\\ 1& 3b& b\\ 1& 4c& c\end{array}\right|=1(3bc-4cb)-2a(c-b)+a(4c-3b)=0$
$\Rightarrow ab+bc=2ac$

$\Rightarrow \frac{b}{c}+\frac{b}{a}=2$

$\Rightarrow \frac{1}{c}+\frac{1}{a}=\frac{2}{b}$

Therefore, $a$, $b$ and $c$ are in H.P.