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).