Solution
$\omega_{1}=(8 \sin \theta+7 \cos \theta)+\mathrm{i}(\sin \theta+4 \cos \theta) ; \omega_{2}=(\sin \theta+4 \cos \theta)+\mathrm{i}(8 \sin \theta+7 \cos \theta)$
\[
\begin{aligned}
& \omega_{1} \omega_{2}=(8 \sin \theta+7 \cos \theta)(\sin \theta+4 \cos \theta)+\mathrm{i}^{2}(\sin \theta+4 \cos \theta)(8 \sin \theta+7 \cos \theta)+\mathrm{i}\left[(\sin \theta+4 \cos \theta)^{2}+(8 \sin \theta+7 \cos \theta)^{2}\right] \\
& \omega_{1} \omega_{2}=0+\mathrm{i}\left(65 \sin ^{2} \theta+120 \sin \theta \cos \theta+65 \cos ^{2} \theta\right) \Rightarrow \omega_{1} \omega_{2}=0+\mathrm{i}(65+60 \sin 2 \theta)=\alpha+\mathrm{i} \beta \therefore \alpha+\beta=0+65+60 \sin 2 \theta
\end{aligned}
\]
We know that $-1 \leq \sin 2 \theta \leq 1 \Rightarrow \mathrm{p}=\alpha+\left.\beta\right|_{\text {max }}=65+60=125 ; \mathrm{q}=\alpha+\left.\beta\right|_{\text {min }}=65-60=5 \Rightarrow \mathrm{p}+\mathrm{q}=125+5=130$
.
