First slide

System of linear equations

Question

$If the system of linear equations x + 2 a y + a z = 0, x + 3 b y + b z = 0 and x + 4 c y + c z = 0 has a non zero solutions; then a, b, c are in \dots \dots$

Moderate

A

A.P
B

G.P
C

H.P
D

Neither A.P nor G.P nor H.P

Solution

$\begin{array}{l} Given equations are x + 2 a y + a z = 0, x + 3 b y + b z = 0, \\ x + 4 c y + c z = 0 \end{array}$
$Since given equation has non-zero solutions \Rightarrow Δ = 0$
$\begin{array}{l} \Rightarrow |\begin{array}{lll} 1 & 2 a & a \\ 1 & 3 b & b \\ 1 & 4 c & c \end{array}| = 0 \\ \Rightarrow 1 (3 b c - 4 b c) - 2 a (c - b) + a (4 c - 3 b) = 0 \\ \Rightarrow \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \end{array}$
$∴ \underset{}{a}, b, c are in Harmonic Progression.$
Therefore, the correct answer is (3).

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