Question:
If \( |z| = 5 \) and \( w = \frac{z-5}{z+5} \), then \( \text{Re}(w) \) is equal to:
If \( |z| = 5 \) and \( w = \frac{z-5}{z+5} \), then \( \text{Re}(w) \) is equal to:
Show Hint
The mapping \( w = \frac{z-k}{z+k} \) maps the circle \( |z|=k \) onto the imaginary axis (where \( \text{Re}(w)=0 \)). This is a fundamental property of Möbius transformations.
Updated On: May 1, 2026
- \( 0 \)
- \( \frac{1}{25} \)
- \( 25 \)
- \( 1 \)
- \( -1 \)
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The Correct Option is A
Solution and Explanation
Concept:
A complex number \( w \) is purely imaginary if: \[ \bar{w} = -w \Rightarrow \text{Re}(w) = 0 \] Also, \[ z\bar{z} = |z|^2 \]
Step 1: Use given modulus.
\[ |z| = 5 \Rightarrow z\bar{z} = 25 \Rightarrow \bar{z} = \frac{25}{z} \]
Step 2: Find conjugate of \( w \).
\[ w = \frac{z-5}{z+5} \Rightarrow \bar{w} = \frac{\bar{z}-5}{\bar{z}+5} \] Substitute \( \bar{z} \): \[ \bar{w} = \frac{\frac{25}{z}-5}{\frac{25}{z}+5} = \frac{25-5z}{25+5z} = \frac{5-z}{5+z} \]
Step 3: Compare with \( w \).
\[ w = \frac{z-5}{z+5} = -\frac{5-z}{5+z} \] Thus, \[ \bar{w} = -w \]
Step 4: Conclusion.
\[ \boxed{\text{Re}(w) = 0} \]
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