standardised random variable least square regression $X$ against $Y$, $Y$ against $X$

Question

Let $X$ and $Y$ be mean 0 and variance 1 random variables such that we choose $\alpha$ and $\beta$ to minimise

$$\mathbb{E}(X-\beta Y)^2$$

and

$$\mathbb{E}(Y-\alpha X)^2$$

after not so difficult derivation, I arrive at $\alpha = \mathbb{E}(XY)/\mathbb{E}X^2$ and $\beta = \mathbb{E}(XY)/\mathbb{E}Y^2$

then $\alpha = \beta$. This seems very strange, because if $y=mx$ is regression line, then surely $x = \frac{1}{m }y$. Why is that or have I calculated $\alpha$ and $\beta$ wrong?

PS: I initialised asked this on cross-validated. The answers I got from there made me want to vomit, see:

https://stats.stackexchange.com/questions/178330/standardised-random-variable-least-square-regression-x-against-y-y-agains

Answer 1

So I will answer my own question. The reason the two regression lines are different is this:

The regression line of $Y$ against $X$ minimises the total squared distance of the $Y$ COORDINATES to the $y$ COORDINATES of the line, where as $X$ against $Y$ minimise the total squared distance of $X$ COORDINATES to the $x$ COORDINATES line.

The confusion I had with regards to the line should be unique comes from the mistaken assumption the regression line minimise the total squared (PERPENDICULAR) distance of each point to the regression line. If this had been the case, my assumption would have been true. However, the two measure of distance never coincide unless there is perfect correlation.