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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?

2good4this
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  • If two points are fixed and the sum of the distances from those points is constant (and greater than the distance between the points) you generally get an ellipse with the two points as the foci. – Mark Bennet Oct 24 '14 at 23:12

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Its an ellipse with focal points $i,-i$.

Troy Woo
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