Complex Numbers

a circle when $\left|z_1-z_2\right|<\lambda$

a parabola when $\left|z_1+z_2\right|=\lambda$

an ellipse when $\left|z_1-z_2\right|<\lambda$

a straight line when $\left|z_1\right|=\left|z_2\right|=\lambda$