Geometry and complex numbers.

Exercise: Describe geometrically, the set of all complex numbers z which satisfy the following condition

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Solution:

Since |z-1| >0, we know the set we want to describe does not contain the point z=1. By the triangle inequality, we have

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for all z. So we want to exclude all points from the plane where the equality

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holds. That is, we want to exclude any z whose distance from 1 is equal to 1 plus its distance to the origin. This just means we have to exclude the negative real axis and the origin. (Draw a picture.)

We can also see this algebraically. Writing z= x+iy we have

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and

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Setting these equal gives

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which reduces to

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This can only hold if tex2html_wrap_inline40 and y=0.

So the set we want is the complex plane with the point z=1 and the segment tex2html_wrap_inline46 deleted.

[Algebra] [Complex Variables]
[Geometry] [Trigonometry ]
[Calculus] [Differential Equations] [Matrix Algebra]

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Author: Michael O'Neill

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