(a) Describe geometrically the curve

$$|\alpha z+\beta \bar{z}|=\sqrt{\alpha \beta}(z+\bar{z})+(\alpha-\beta)^{2},$$

where $z \in \mathbb{C}$ and $\alpha, \beta$ are positive, distinct, real constants.

(b) Let $\theta$ be a real number not equal to an integer multiple of $2 \pi$. Show that

$$\sum_{m=1}^{N} \sin (m \theta)=\frac{\sin \theta+\sin (N \theta)-\sin (N \theta+\theta)}{2(1-\cos \theta)}$$

and derive a similar expression for $\sum_{m=1}^{N} \cos (m \theta)$.