Simple linear regression without intercept - Expected Value and Variance Estimator (Slope)

Question

I'm solving an exercise while studying for exam, I have been asked to find the estimator of simple linear model without intercept estimator, its expected value and variance.

I got for the Estimator B₁ = Sum(x_i.y_i)/sum(x_i)²

Which I think is correct.

For Expected value I think E(B₁)= E(Sum(x_i.y_i)/sum(x_i)²)=B₁

Because numerator is constant and denominator is definitely constant, no?

Estimator's Variance I got:

1/(sum(x_i)²)² * var(Sum(x_i.y_i)) which is constant so, Sum(x_i.y_i)/(sum(x_i)²)²

If this isn't right then for y_i should I substitute for the model I got with the estimator and see what i get?