Please verify whether the calculating process of the WLS estimator and the variance is correct or not.

Question

There are two Heteroscedasticity regression models

1.

$$ y_i = \beta x_i + \epsilon_i, \quad i=1, \ldots, n $$ where $\epsilon_i$'s are independent and distributed as $\epsilon_i \sim N(0, \sigma^2 x_i^2)$

Find weighted least squares estimator for β and its variance.

Process 1

2.

$$ y_i = \beta x_i + \epsilon_i, \quad i=1, \ldots, n $$ where $x_i$>0 , E($\epsilon_i$)=0 , but Var($\epsilon_i$)=$\sigma^2 x_i$

Find the best linear unbiased estimator (BLUE) of β and its variance.

Process 2

I am not sure whether the processes are correct or not.

Please help me to verify whether the calculating process for the two questions above are correct.

Please also tell me why we can not directly use OLS methods in question 1, but have to weight them first.

Thank you.