Suppose that $Y_1,Y_2,\ldots,Y_n$ are independent $\mathcal{N}(\alpha x_i,\sigma^2)$ variates where $x_1,x_2,\ldots,x_n$ are known constants. Show that the likelihood ratio statistics for testing a value of $\alpha$ is given by $D=n \log (1+\left(\frac{1}{n-1} T^2\right))$, where $T=(\hat{\alpha}{−\alpha})/\sqrt{s^2 \cdot c}$ where $c=\left(\sum_{i} x_i^2\right)^{-1}$.
Hint: you will need to show that $\sum_i (y_i{-\alpha} x_i)^2 = \sum (y_i - \hat{\alpha}{x_i})^2+(\hat{\alpha}{-\alpha})^2 \sum(x_i^2)$