A) Satisfy \[a+2b+3c=0\]
B) Are in A.P
C) Are in G.P
D) Are in H.P.
Correct Answer: D
Solution :
[d] For homogeneous system of equations to have non zero solution, \[\Delta =0\] \[\left| \begin{matrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & ac & c \\ \end{matrix} \right|=0[\therefore {{C}_{2}}\to {{C}_{2}}-2{{C}_{3}}]\] \[\left| \begin{matrix} 1 & 0 & a \\ 1 & b & b \\ 1 & 2c & c \\ \end{matrix} \right|=0\,[{{R}_{3}}\to {{R}_{3}}-{{R}_{2}},\,\,{{R}_{2}}\to {{R}_{2}}-{{R}_{1}}]\] On simplification, \[\frac{2}{b}=\frac{1}{a}+\frac{1}{c}\] \[\therefore \] a, b, c are in Harmonic Progression.
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