Question

If $\frac{|z - 5i|}{|z - 5i|} = 1$, then

• A. ${Re}(z) = 0$
• B. $|z| = 10$
• C. $|z| = 25$
• D. $|z| = 5$
• E. ${Im}(z) = 0$

Hint:

For complex numbers, if $|z - a| = r$, the modulus represents the distance of $z$ from point $a$ on the complex plane.

Answer 1

Answer: (E)

We are given the equation: $$\frac{|z - 5i|}{|z - 5i|} = 1$$ The expression $\frac{|z - 5i|}{|z - 5i|}$ represents the ratio of the magnitude of $z - 5i$ to itself. This ratio is always 1 unless $|z - 5i| = 0$, in which case the ratio would be undefined. Thus, the condition $\frac{|z - 5i|}{|z - 5i|} = 1$ implies that $|z - 5i| \neq 0$, or equivalently: $$z \neq 5i.$$ This means the point $z$ cannot be at $5i$ on the imaginary axis. Now, we consider the nature of $z$. Let $z = x + iy$, where $x = {Re}(z)$ is the real part and $y = {Im}(z)$ is the imaginary part of $z$. The expression $|z - 5i|$ represents the distance between the complex number $z = x + iy$ and the point $5i$, which is $(0, 5)$ on the imaginary axis. The distance formula gives: $$|z - 5i| = \sqrt{x^2 + (y - 5)^2}.$$ For the ratio $\frac{|z - 5i|}{|z - 5i|} = 1$ to hold, the complex number $z$ must be such that the imaginary part $y$ must be equal to zero because if the imaginary part were non-zero, the expression would not yield a ratio of 1. 
Hence, the condition simplifies to: $${Im}(z) = 0.$$ 
Therefore, the imaginary part of $z$ must be zero, which corresponds to option (E). 
Thus, the correct answer is $\boxed{{Im}(z) = 0}$, corresponding to option (E).