Question

If the following three linear equations have a non-trivial solution, then
$x+4ay+az=0$
$x+3by+bz=0$
$x+2cy+cz=0$

• A. a, b, c are in A.P.
• B. a, b, c are in G.P
• C. a, b, c are in H.P.
• D. a + b + c = 0

Answer 1

Answer: (C)

To have non-trivial solution, $\left|\begin{array}{ccc}1& 4a& a\\ 1& 3b& b\\ 1& 2c& c\end{array}\right|=0$
$\Rightarrow 1(3bc-2bc)-(4ac-2ac)+1(4ab-3ab)=0$
$\Rightarrow bc-2ac+ab=0$
$\Rightarrow bc+ab=2ac$
$\Rightarrow b(a+c)=2ac$
$\Rightarrow b=\frac{2ac}{a+c}$
$\Rightarrow a,b,c$ are in $HP$