Conditional expectation of sum of random variables

Question

x and y are independent random variables:

$x \sim Norm(0, \sigma_1^2)$

$y \sim Norm(0, \sigma_2^2)$

It is known that $x + y = \alpha$ How to find $E(x|\alpha)$ ?

Answer 1

$$E[x|\alpha] = \alpha \dfrac{\sigma_1^2}{\sigma_1^2+\sigma_2^2}$$

$$Var(x|\alpha) = \dfrac{\sigma_1^2 \sigma_2^2}{\sigma_1^2 + \sigma_2^2}$$

See both the question and answer at https://stats.stackexchange.com/questions/9071/intuitive-explanation-of-contribution-to-sum-of-two-normally-distributed-random for an explanation