Simplifying Likelihood Ratio

Question

(Answered here: https://stats.stackexchange.com/questions/372040/rejection-region-for-likelihood-ratio-test)
I have a data set $((Y_1,x_1),(Y_2,x_2),...,(Y_n,x_n))$ where $Y_i$ is distributed as $N(\theta x_i,1)$. I want to perform a likelihood ratio test for $\theta$ to investigate the hypothesis $H_0: \theta=\theta_0$ or $H_1: \theta\neq\theta_0$: $$\lambda(X)=\frac{exp(\frac{-\sum_i(Y_i-x_i\theta_0)^2}{2})}{exp(\frac{-\sum_i(Y_i-x_i\hat\theta)^2}{2})}$$ Where $\hat\theta$ is the maximum likelihood estimator, which I have determined to be $\frac{\sum_i x_iY_i}{\sum_ix_i^2}$. I want to simplify this ratio as much as possible, but I don't know how to go about this, because $x_i\theta$ varies with $i$, and as such, cannot be taken out of the sum. Normally, I would solve this by adding $0$ (i.e. adding and subtracting a term), but this approach did not help me this time around. Is there a trick that I am missing? Thanks in advance.