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Let \(z\) be a complex number such that \(|z - 1| \le 3.\) Find all possible values of \(|iz + 3 - 5i|.\) Enter your answer as an interval.

 Feb 18, 2021
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Let z = x + yi, so |z - 1| = |(x - 1) + yi| = sqrt((x - 1)^2 + y^2) <= 3, or (x - 1)^2 + y^2 <= 9.

 

Also, |iz + 3 - 5i| = |i(x + yI) + 3 - 5i| = |ix - y + 3 - 5i| = sqrt((-y + 3)^2 + (x - 5)^2).

 

Taking the derivative of (x - 1)^2 + y^2 <= 9 gives us (2(x - 1), 2y).  We then set this to to the derivative (-y + 3)^2 + (x - 5)^2, including a multiplier: (2k(-y + 3), 2k(x - 5)).

 

Solving, we get that the minimum of |iz + 3 - 5i| is 3 and the maximum is 7, so the set of all possible values is [3,7].

 Feb 18, 2021

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