Consider the model $y_i=\beta x_i + \epsilon_i$ (without a constant term and with $k=1$), where $\mathbb{E}[\epsilon_i]=0, \mathbb{E}[\epsilon_i \epsilon_j]=0, \forall i \neq j$, and $\mathbb{E}[\epsilon_i^2]=\sigma_i^2$.
Then consider following estimators of $\beta:$
$$b_1=\frac{\sum{x_iy_i}}{\sum{x_i^2}},$$
$$b_2=\frac{\sum y_i}{\sum x_i}$$
and derive a model for the variances $\sigma_i^2$ for which this estimator is the best linear unbiased estimator (BLUE) of $\beta$
My attempt so far: I thought I should write this model in the 'standard form' for which we know that the OLS estimator is BLUE. So I tried to deduce the variance of this estimator so that maybe this could be corrected by a certain factor (similarly to in Weighted Least Squares).
The variance that I got is:
$$\mathrm{Var}[b_1]=\frac{\sum x_i^2 \sigma_i^2}{\left(\sum x_i^2\right)^2}.$$
However, it doesn't seem that I can correct this variance by a factor to bring it to that of the standard model. Same problem for $b_2$. Could anyone please help?