Locus of Complex Number

Question

Would be great to get your help in finding the locus of this complex number $z$: $|z-z_1|+\sin \alpha|z-z_2|=\sin \theta$

From this question I proceed to a refined one-
What would $$|z-z_1|+2|z-z_2|=k$$
represent?

Answer 1

Here's a hint. If $\alpha = \pi / 2$, then this equation says that the sum of the distances from $z$ to $z_1$ and $z_2$ is a fixed constant -- i.e. $z$ is on an ellipse with foci at $z_1$ and $z_2$. What happens if $\alpha=0$ or $\alpha = \pi$? Can you generalize?