Let z be a complex number such that |z−1|≤3. Find all possible values of |iz+3−5i|. Enter your answer as an interval.
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].
