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 November 10th, 2010, 11:59 AM #1 Senior Member   Joined: Sep 2009 Posts: 251 Thanks: 0 Where's the center of this circle? Stupid question here (I assume): C: |z+1/4|=1/2 is a circle of radius 1/2 centered at 1/2 on the complex plane. Right? Why is someone saying it's centered at -1/2 on the plane. If z=0, z+/4 is 1/4. Then the radius should stretch 1/2 in all directions from there. Right? Thanks.
 November 10th, 2010, 02:15 PM #2 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: Where's the center of this circle? [color=#000000]The circle of center $\bf z_{0}$ and radius $\bf R$ on the complex plain is $\bf |z-z_{0}|=R$ . So for your example $\bf \left|z-\left(-\frac{1}{4}\right)\right|=\frac{1}{2}$, which means that the circle is centered at $\bf\left(-\frac{1}{4},0\right)$ and has radius $\bf\frac{1}{2}$.[/color]
November 10th, 2010, 03:18 PM   #3
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Re: Where's the center of this circle?

Quote:
 Originally Posted by ZardoZ [color=#000000]The circle of center $\bf z_{0}$ and radius $\bf R$ on the complex plain is $\bf |z-z_{0}|=R$ . So for your example $\bf \left|z-\left(-\frac{1}{4}\right)\right|=\frac{1}{2}$, which means that the circle is centered at $\bf\left(-\frac{1}{4},0\right)$ and has radius $\bf\frac{1}{2}$.[/color]
OK, I can follow that easy enough. But what I guess I'm really asking is "why"? Why is the center at (-1/4,0) (or why is it at -z_0 in the general equation)?

Isn't the center going to be when z=0? And when z=0 for the circle |z+1/4|=1/2, the point is (0,1/4). So the center should be (0,1/4), and z will circle (0,1/4) at a distance of |1/2|? Put another way, the magnitude of z from 1/4 will always be 1/2. 1/2 unit away the point 1/4. Either the theorem you quote is wrong (it's not!) or my thought process is wrong. But I can't see what's wrong with my thinking for the life of me.

 November 11th, 2010, 02:59 AM #4 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,989 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: Where's the center of this circle? [color=#000000]When you have an equation which represents a circle on the complex plain, this means that you are talking about the points that are on the circle, not inside the circle. Above you took $\bf z=0$ which means that it is an internal point of the circle not a boundary one which is the correct. What you say though could be right , if$\bf |z-z_{0}|\leq R$ which represents the points that are interior points plus the boundary ones.[/color]

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