Complex Trouble 1

Algebra Level 4

What is the maximum value of z 2 2 i z + 1 |z^2 - 2iz+1| given that z = 3 |z|=3 ?


The answer is 14.

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1 solution

So by simplifying the given expression,

= z 2 2 i z + 1 |z^{2} - 2iz + 1|

= z 2 2 i z + 1 + 1 1 |z^{2} - 2iz +1 + 1 - 1|

= z 2 2 i z 1 + 2 |z^{2} - 2iz - 1 + 2|

= ( z i ) 2 + 2 |(z-i)^2 + 2|

As they have given that z = 3 |z| = 3 , it implies that z lies on a circle of radius 3.

So the distance between z and i must be maximum.

Therefore z = 3 i z = -3i .Substituting this in the given expression we get 14 14 .

Nice problem!

Harsh Shrivastava - 4 years, 3 months ago

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