I have been asked to describe the subset of the complex plane which is formed by the complex numbers z satisfying |z-i| + |z+i| = 3.
It was easy to see that if the points z lie on the line segment joining the points i and -i, then z can only be at a distance of 1/2 above i or 1/2 below -i. But for z not lying on that line segment, I tried using the cosine rule to find out in what range the points z should be, but this did not seem very fruitful.
Any hints towards a better analysis of this situation? I have a feeling the points I seek must lie on an ellipse with foci at i and -i, but is there a way to solve this problem without having to deduce an equation of an ellipse?
Its an ellipse with focal points $i,-i$.