In the equation $|z-1|^n=k$ one of the roots is $3+4i$

Question

In the equation $|z-1|^n=k$ (where n belongs to the set of natural numbers and k to the set of real numbers) has a root $4+3i$,then:

A) A possible value of n is 60.

B) A possible value of n is 200.

C)The least possible value of|k| lies in (0,100)

D)The least possible value of |k| lies in (300,400)

Do we solving this by taking $z=x+iy$ ,if so its not yielding any result but complicating it further with making it an equation in 3 variables.

Also a geometric approach to this problem will be greatly appreciated.

Answer 1

Substituting $\,z=3+4i\,$ in the equation gives $\,k = |2+4i|^n=\left(2\sqrt{5}\right)^n\,$. Since $2\sqrt{5} \gt 1$ it follows that $\,k = \left(2\sqrt{5}\right)^n \ge 2\sqrt{5}\,$ with equality iff $\,n=1\,$, so C) is true and D) is false. As for the first two points, each natural value of $\,n\,$ gives a real $\,k\,$, so $\,\ldots$