B̂xy and B̂yx are slopes in regressions of x on y and of y on x respectively. Then, the correlation…

In linear regression between two variables $x$ and $y$:

The slope of the regression line of $x$ on $y$ is $\hat{B}_{xy}=\frac{\operatorname{Cov}(x,y)}{\operatorname{Var}(y)}=r\cdot\frac{\sigma_x}{\sigma_y}$.

The slope of the regression line of $y$ on $x$ is $\hat{B}_{yx}=\frac{\operatorname{Cov}(x,y)}{\operatorname{Var}(x)}=r\cdot\frac{\sigma_y}{\sigma_x}$. Multiplying the two regression slopes together cancels the standard deviations:

$\hat{B}_{xy}\cdot\hat{B}_{yx}=r^2\Rightarrow |r|=\sqrt{\hat{B}_{xy}\cdot\hat{B}_{yx}}$. Thus, the correlation coefficient $r$ is the geometric mean of the two regression slope coefficients (with the same algebraic sign as the two slopes).

Revised question: the original examination question was defective.