If the system of linear equations

Question:

If the system of linear equations

$2 x+2 a y+a z=0$

$2 x+3 b y+b z=0$

$2 x+4 c y+c z=0$

where $a, b, c \in \boldsymbol{R}$ are non-zero and distinct; has a non-zero solution, then:

  1. (1) $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.

  2. (2) $a, b, c$ are in G.P.

  3. (3) $a+b+c=0$

  4. (4) $a, b, c$ are in A.P.


Correct Option: 1

Solution:

For non-zero solution

$\left|\begin{array}{lll}2 & 2 a & a \\ 2 & 3 b & b \\ 2 & 4 c & c\end{array}\right|=0$

$\Rightarrow\left|\begin{array}{lll}1 & 2 a & a \\ 1 & 3 b & b \\ 1 & 4 c & c\end{array}\right|=0$

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

$\Rightarrow-b c+2 a c-a b=0$

$\Rightarrow a b+b c+2 a c$

$\Rightarrow \quad \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$

$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ in A.P.

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