Two regression slopes

Idea. Each slope is the covariance divided by the variance of the regressor. In the product, the covariance appears twice and each variance once, which leaves the squared correlation.

Step 1: the two slopes. With ${\sigma }_X=2$, ${\sigma }_Y=5$, $\rho =0.6$, $${\beta }_{Y|X}=\frac{\mathrm{Cov}(X,Y)}{{\sigma }_X^2}=\rho \frac{{\sigma }_Y}{{\sigma }_X}=0.6\cdot \frac{5}{2}=1.5,$$ $${\beta }_{X|Y}=\frac{\mathrm{Cov}(X,Y)}{{\sigma }_Y^2}=\rho \frac{{\sigma }_X}{{\sigma }_Y}=0.6\cdot \frac{2}{5}=\mathrm{0.24.}$$

Step 2: product. $${\beta }_{Y|X}\cdot {\beta }_{X|Y}={\rho }^2=0.36=\frac{9}{25}.$$ The standard deviations were not needed.

Why it matters. If you draw both regression lines in the $(x,y)$ plane, the second one has slope $\frac{1}{{\beta }_{X|Y}}=\frac{1}{0.24}\approx 4.17$ in that plane, steeper than $1.5$. The two lines coincide only when $|\rho |=1$. Regressing $X$ on $Y$ and inverting the slope does not give the regression of $Y$ on $X$: both regressions shrink toward the mean.