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# For two variables $x$ and $y$ with the same mean the two regression lines are $x+2y-5=0$ $2x+3y-8=0$ and variance $(x)=12$ Then $\sigma_y$ is

$\begin {array} {1 1} (A)\;4 & \quad (B)\;5 \\ (C)\;2 & \quad (D)\;None\: of \: these \end {array}$

Slopes of regression lines are $-\large\frac{1}{2}$ and $\large\frac{-2}{3}$
$\therefore b_{yx} = -\large\frac{ 1}{2}$ and $b_{xy}= -\large\frac{3}{2}$
$\therefore r^2 = \bigg( -\large\frac{1}{2} \bigg) \bigg( -\large\frac{3}{2} \bigg) = \large\frac{3}{4}$ $( < 1)$
$\Rightarrow r = -\large\frac{\sqrt 3}{2}$
Also $b_{yx}=r\large\frac{\sigma_y}{\sigma_x}$
$\sigma_x=2\sqrt 3$
$\therefore \sigma_x^2 = 12$
$-\large\frac{1}{2}$ $= -\large\frac{\sqrt 3}{2}.\large\frac{\sigma_y}{2\sqrt 3}$
$\sigma_y = 2$
Hence Ans (C)