Question

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

• A. $A.P.$
• B. $G.P.$
• C. $H.P.$
• D. $satisfiesa+2b+3c=0$

Answer 1

Answer: (C)

$H.P.$

$\left|\begin{array}{ccc}1& 0& a\\ 1& b& b\\ 1& 2c& c\end{array}\right|=0$

On expanding along $R_1$ we get

$1(bc-2bc)+0(c-b)+a(2c-b)=0$

$\Rightarrow 2ac=ab+bc$

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

$\Rightarrow a,b,c$ are in H.P