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# complex numbers help Watch

1. i can't think how to find the locus of points such that |z-1|=|z-3|=2

i've tried writing |z-1|=2-|z-3|=|2-(z-3| =|1-z| and i get the same thing taking |z-3| over to the other side. i've tried a number of variations on this too, but nothing is working... any help would be much appreciated!
2. Consider the locus of points where |z-1|=2 and where |z-3|=2. These form two circles, one centred on 1 with a radius of 2, and one centred on 3 with a radius of 2. If the condition was just |z-1|=|z-3|, then there would exist a vertical line between 1 and 3 (perpendicular bisector of the line segment from 1 to 3) where there are an infinite number of points where this condition is met. However, since they both equal 2, there are only two points which satisfy the condition, and these are where the two circles meet.

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Updated: December 1, 2010
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