Question

If three linear equations x + 4ay + az = 0, x + 3by + bz = 0 and x + 2cy + cz = 0 have a non-trivial solution, then a, b, c are in-

• A. H.P
• B. G.P.
• C. A.P.
• D. None

Answer 1

Answer: (A)

H.P

Answer 2

∆ = 0 (Q non trivial solution )

⇒ $\left| \begin{matrix} 1 & 4a & a \\ 1 & 3b & b \\ 1 & 2c & c \end{matrix} \right|$ = 0

⇒ 1(3bc – 2bc) – 4a(c – b) + a(2c – 3b) = 0

⇒ bc – 4ac + 4ab + 2ac – 3ab = 0 ⇒ bc – 2ac + ab = 0

⇒ bc + ab = 2ac

⇒ $\frac{1}{a}$ + $\frac{1}{c}$ = $\frac{2}{b}$ ∴ a, b, c are in H.P