Value of r, i.e. correlation coefficient, of data points on a horizontal straight line 
*

*If we use the formula: 
$$r = \displaystyle \frac{\displaystyle\sum_{i=1}^n \left( x_i - \bar{x} \right) \left( y_i - \bar{y} \right)}{\sqrt{\displaystyle \sum_{i=1}^n \left( x_i - \bar{x}\right)^2 \cdot \sum_{i=1}^n \left( y_i - \bar{y}\right)^2}}$$ 
for calculating $r$, we get $r = \frac{0}{0}$. 

*If we see $r$ as a measurement of the strength and nature of the linear relationship between two variables, we get $r = 0$ because there is no linear relationship in this case.

*If a perfect positive fit gives $r = 1$, and a perfect negative fit gives $r = -1$, then it seems like we should get $r = 1$ or $-1$. (I think this is wrong though.)


Which is correct?

 A: From the definition below:
$r = \displaystyle \frac{\displaystyle\sum_{i=1}^n \left( x_i - \bar{x} \right) \left( y_i - \bar{y} \right)}{\sqrt{\displaystyle \sum_{i=1}^n \left( x_i - \bar{x}\right)^2 \cdot \sum_{i=1}^n \left( y_i - \bar{y}\right)^2}}$, or equally $r = \displaystyle \frac{1}{n-1} \sum_{i=1}^n \left( \frac{x_i - \bar{x}}{s_x}\right) \left( \frac{y_i - \bar{y}}{s_y}\right)$
You get answer one which seems correct to me since you have identical values of $y$ independently of the different $x_i$. And since $y_i = \bar{y}$ for all $i$, you get $0$'s in column ${\tt K}$ of your table and end up with $r = \frac{0}{0}$. 
Answer two is discarded because $s_y = \displaystyle \sqrt{\frac{1}{n-1}\sum_{i=1}^n \left(y_i - \bar{y}\right)^2}=0$ as well.
You would get answer three if the variations $\Delta x_i$ and $\Delta y_i$ were at least directly or indirectly proportional. Which isn't the case here. So, $r > 0$ and $r < 0$ are discarded too.
You will still get the same answer one, $r = \frac{0}{0}$ if you switch $x$ and $y$ values. You might want to check this question on correlation coefficients on random variables. The answers and comments are great. 
