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Complex condition (Posted on 2024-06-05)

Let z be complex number such that |z+2/z| =2. Find maximum value of |z|.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

| Comment 1 of 2

Let z=(r,t) be the polar form. Then abs[z] = r.

Then abs[(r,t)+2/(r,t)] = 2

abs[(r,t)+(2/r,-t)] = 2

abs[(r,t)-(2/r,t)] = 2

abs[((r-2/r),t)] = 2

abs[r-2/r] = 2

r-2/r = +/-2

r-2/r = 2 -OR- r-2/r = -2

r^2-2=2r -OR- r^2-2=-2r

r^2-2r-2=0 -OR- r^2+2r-2=0

r=-1+/-sqrt(3) -OR- r=-1+/-sqrt(3)

The largest of these is r=1+sqrt[3].

Posted by Brian Smith on 2024-06-05 13:58:16

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