If $y = m_{1} x + c_{1}$ is the regression line of $y$ on $x$ and $y = m_{2} x + c_{2}$ is the regression line of $x$ on $y$ , then which is true $?$

$m_{1} m_{2} < 1$
$0 \leq \sqrt{m_{1}} m_{2} \leq 1$
$- 1 \leq \sqrt{\frac{m_{1}}{m_{2}}}$
$- 1 \leq \sqrt{\frac{m_{2}}{m_{1}}}$

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Solution

The correct option is C
$- 1 \leq \sqrt{\frac{m_{1}}{m_{2}}}$

Step 1: Find the regression coefficient of both the lines
We know that the regression coefficient of $y$ on $x$ indicates the change in the value of a variable $y$ corresponding to a unit change in the value of the variable $x$
Given that, $y = m_{1} x + c_{1}$ is the regression line of $y$ on $x$
$∴ b_{y x} =$ The regression coefficient of $y$ on $x$ $= m_{1}$ $. . . . (1)$
Where $m_{1} =$ The slope of the line $y = m_{1} x + c_{1}$ , which is the change in the value of a variable $y$ corresponding to a unit change in the value of the variable $x$
Now, $y = m_{2} x + c_{2}$ is the regression line of $x$ on $y$ ,
$\Rightarrow$ $x = \frac{1}{m_{2}} y - \frac{c_{2}}{m_{2}}$ is the regression line of $x$ on $y$ ,
$∴ b_{x y} =$ The regression coefficient of $x$ on $y$ $= \frac{1}{m_{2}}$ $. . . . (2)$
Where $m_{2} =$ The slope of the line $y = m_{2} x + c_{2}$ , which is the change in the value of a variable $x$ corresponding to a unit change in the value of the variable $y$
Step 2: Use the range of correlation coefficient
The correlation coefficient is the geometric mean of the regression coefficient.
i.e., $r = \sqrt{b_{x y} \times b_{y x}}$
We know that the range of correlation coefficient is $- 1$ to $1$ . i.e., $- 1 \leq r \leq 1$
$\Rightarrow - 1 \leq \sqrt{b_{x y} \times b_{y x}} \leq 1$
$\Rightarrow - 1 \leq \sqrt{\frac{1}{m_{2}} \times m_{1}} \leq 1$ [Using $(1)$ and $(2)$ ]
$\Rightarrow - 1 \leq \sqrt{\frac{m_{1}}{m_{2}}}$
As per the given options, we can observe that option C is the only one that satisfies the condition.
Hence, the correct option is option C.

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