Question

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

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

Answer 1

The correct option is B H.P.

For non trivial solution,$\Delta =0$

$i.e.\left|\begin{array}{ccc}1& 2a& a\\ 1& 3b& b\\ 1& 4c& c\end{array}\right|=0$

$\Rightarrow (3bc-4bc)-2a(c-b)+a(4c-3b)=0$

$\Rightarrow \frac{2ac}{a+c}=b\Rightarrow a,b,c$ are in H.P.

Hence, option B.