If $4x-5y+33=0$ and $20x-9y-107=0$ are two lines of regression. Find the regression coefficient, ${{b}_{xy}}$.
(a) $\dfrac{1}{20}$
(b) $\dfrac{1}{10}$
(c) $\dfrac{1}{30}$
(d) $\dfrac{9}{20}$

Hint: First, we will write down the given data, which are the two regression lines. Next step is to choose one of the equations and simplify it further to the general form of the regression line where X depends on Y and compare with it and find the coefficient of regression, ${{b}_{xy}}$.

Complete step by step answer:
Here, we have been given two lines of regression, $4x-5y+33=0$ and $20x-9y-107=0$. We will be using $20x-9y-107=0$ to find the regression coefficient, ${{b}_{xy}}$.
Regression coefficient, ${{b}_{xy}}$ means that the $x$ variable depends on $y$ variable.
Let us suppose $20x-9y-107=0$ is a regression equation of $x$ on $y$.
Let us first add 107 on both the sides of the equation, we get
$20x-9y-107+107=0+107$
Now, let us solve further, we get
$20x-9y=107$
In the next step, let us add $9y$ on both the sides of the equation, we get
$20x-9y+9y=9y+107$
After solving it further, we will get
$20x=9y+107$.
In the final step, we will divide by 20 throughout the equation and get the value of the variable $x$. Therefore, after dividing, we get
$\dfrac{20x}{20}=\dfrac{9y+107}{20}$
Now, let us solve it further to get the value of $x$, we get
$x=\dfrac{9}{20}y+\dfrac{107}{20}$
Now, we know the general form of the regression line X = A + BY, where B is the regression coefficient, ${{b}_{xy}}$.
Therefore, when we compare the obtained equation and the general form, we will get the regression coefficient.
We get,
${{b}_{xy}}=\dfrac{9}{20}$.

Hence, the required regression coefficient is $\dfrac{9}{20}$.

Note: Here, in this question, we took the second equation because of the options in the question, we could have taken the other equation as well, we would have got a different answer which would be correct but not the required answer from the options. If X depends on Y, then the regression line is X on Y and X is dependent variable and Y is independent variable.