माना \(z \neq-i\) कोई ऐसी सम्मिश्र संख्या है कि \(\frac{z-i}{z+i}\) एक शुद्ध काल्पनिक संख्या है, तो \(z+\frac{1}{z}\) है

A \(0\)
B \(1\) के अतिरिक्त कोई शून्येत्तर वास्तविक संख्या।
C कोई शून्येत्तर वास्तविक संख्या।
D एक शुद्ध काल्पनिक संख्या।

Verified Solution

Answer & Solution

Correct Answer

(C) कोई शून्येत्तर वास्तविक संख्या।

Step-by-step Solution

Detailed explanation

Let \(z=x+i y\) \(\frac{z-i}{z+i}\) is purely imaginary means its real part is zero. \(\frac{x+i y-i}{x+i y+i}=\frac{x+i(y-1)}{x+i(y+1)} \times \frac{x-i(y+1)}{x-i(y+1)}\) \({ = \frac{{{x^2} - 2ix(y + 1) + xi(y - 1) + {y^2} - 1}}{{{x^2} + {{(y + 1)}^2}}}}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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