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If $$|z-(3+4\iota)|\le 3$$ find the complex number with least magnitude satisfying the above inequality.

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I recognized that the inequality represents the interior and circumference of a circle set in the Argand Plane with radius 3, centered at $(3,4)$. This is unfortunately as far as I got with the problem. $$$$ A friend of mine mentioned that z of the least magnitude is where the line joining the Origin to the center of the circle meets the circle's circumference. This seemed a very arbitrary declaration to me for I couldn't understand why this is so. I've tried to prove this proposition, but have so far been unable to. $$$$ I would greatly appreciate any help with this problem. While all methods are welcome, a method using the Rotation Theorem is preferred as this problem was initially part of a series of practice problems on the Rotation Theorem.

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  • $\begingroup$ Your friend's answer is intuitive. You need to find the point on this circle that is the closest to the origin, hence having the least magnitude. This point is the intersection between the circumference and the line connecting the origin and the center of the circle. $\endgroup$ – user164550 Mar 24 '16 at 19:09
  • $\begingroup$ Also by intuition, you can tell that is has a magnitude of $2$ and angle of $\arctan(\frac{4}{3})$. $\endgroup$ – user164550 Mar 24 '16 at 19:12
  • $\begingroup$ Also, you can think of it this way, if you rotate the whole plane by the aforementioned angle, the equation would be $|z'-5|\leq3$. This means the point is $2+0i$, then you rotate back with the same angle and you will get the same answer. $\endgroup$ – user164550 Mar 24 '16 at 19:14
  • $\begingroup$ I understand that it has to be closest to the origin so as to have the least possible magnitude. But I couldn't understand why it lies on the line connecting the origin and the center of the circle. Could you please give a more detailed answer? Thanks in advance! $\endgroup$ – Better World Mar 24 '16 at 19:15
  • $\begingroup$ Sorry, but I didn't get the part ' Also by intuition, you can tell that is has a magnitude of 2 and angle of $\arctan(4/3)$.'. Could you please elaborate? Thanks. $\endgroup$ – Better World Mar 24 '16 at 19:16
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That's great you realized that this inequality represents all the points in the interior and on the circumference of the circle centered at (3,4) with radius 3.

If you know the basics I suggest you directly skip to point 2.

  1. Understanding the modulus of a complex number

If $z=x+iy$, then $|z|=\sqrt{x^2+y^2}$; which effectively represents the distance of the point (x,y) from the origin.

When we talk about $|z-z'|$ (where z' is another complex number) we represent the distance of complex number z from z' on the Argand plane. So, the inequality that you encountered represents any variable complex number z that has 3(or less than 3) unit distance from z' (which in your case is 3+4i or (3,4) point on the argand plane), and thus it represents a circle.

  1. Coming to the question, draw a rough sketch of a circle(centered (3,4);radius 3) on a paper having relative co-ordinate system and then you would instantly realize that the value of the least modulus(distance from origin) of the variable complex number z lies on the circumference of the circle. It will be lying on the line joining the origin and the center of the circle. And, in this case it is 2.
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