Pearsons corelation coefficient is defined as follows

$$ r_{x,y} =frac {sum (x_i -bar X)(y_i – bar Y)}{sqrt{sum(x_i-bar X)^2} sqrt{sum(y_i-bar Y)^2}} $$.

Now, maximum magnitude of a correlation coefficient is known to be $1$;

which is possible only if

$$Numarator = denominator$$,

$$or, {sum (x_i -bar X)(y_i – bar Y)}={sqrt{sum(x_i-bar X)^2} sqrt{sum(y_i-bar Y)^2}} … (1) $$;

Now; if the dataset is **perfectly corelated**, then how or why the relation $… (1)$ works? or in other words, the denominator or numarator becomes equal?

Samely the alternate equation,

$$r_{xy}= frac {nsum X_iY_i- (sum x_i)(sum y_i)} {sqrt{nsum x_i^2 -(sum x_i)^2} sqrt{nsum y_i^2-(sum y_i)^2}}$$

now, the coefficient of correlation will be =1 only if

$$numerator=denominator$$

$$or,{nsum X_iY_i- (sum x_i)(sum y_i)} = {sqrt{nsum x_i^2 -(sum x_i)^2} sqrt{nsum y_i^2-(sum y_i)^2}} … (2) $$

If the dataset is perfectly correlated ; then how to tell relation $… (2)$ will work?

So in brief my question is ….

I am looking for necessarily intuitive explanation of what these mathematical terms mean; such as it is quite clear that ${sum (x_i -bar X)(y_i – bar Y)}$ is a sum of **areas** made by the data points which comes from covariance. **But why this exact same area will be cancelled out by the denominator**, that I could not understand by any way.

What I tried already:

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