If z be any complex number `(z!=0)` then `arg((z-i)/(z+i))=pi/2`represents the curve

Options

A. `|z|=1`
B. `|z|=1, "Re"(z) gt 0`
C. `|z|=1, "Re"(z) lt 0`
D. none of these

Explanation

To solve the problem, we need to analyze the expression given and find out what the condition `arg((z-i)/(z+i)) = pi/2` represents in terms of a curve.

### Step-by-step Solution:

1. **Understanding the Expression**:
We start with the expression:
\[ \arg\left(\frac{z - i}{z + i}\right) = \frac{\pi}{2} \]
This means that the argument of the complex number \(\frac{z - i}{z + i}\) is \(\frac{\pi}{2}\).

2. **Using the Property of Argument**:
We can use the property of the argument of a quotient:
\[ \arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2) \]
Thus, we can rewrite our expression as:
\[ \arg(z - i) - \arg(z + i) = \frac{\pi}{2} \]

3. **Substituting \( z \)**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then:
\[ z - i = x + (y - 1)i \]
\[ z + i = x + (y + 1)i \]

4. **Finding the Arguments**:
The arguments can be computed as:
\[ \arg(z - i) = \tan^{-1}\left(\frac{y - 1}{x}\right) \]
\[ \arg(z + i) = \tan^{-1}\left(\frac{y + 1}{x}\right) \]

5. **Setting Up the Equation**:
Substituting these into our earlier equation gives:
\[ \tan^{-1}\left(\frac{y - 1}{x}\right) - \tan^{-1}\left(\frac{y + 1}{x}\right) = \frac{\pi}{2} \]

6. **Using the Tangent Difference Formula**:
We can use the tangent difference formula:
\[ \tan\left(\tan^{-1}(A) - \tan^{-1}(B)\right) = \frac{A - B}{1 + AB} \]
Setting \( A = \frac{y - 1}{x} \) and \( B = \frac{y + 1}{x} \), we have:
\[ \tan\left(\frac{\pi}{2}\right) = \infty \]
This implies that the denominator must be zero:
\[ 1 + \left(\frac{y - 1}{x}\right)\left(\frac{y + 1}{x}\right) = 0 \]

7. **Simplifying the Denominator**:
Simplifying the denominator gives:
\[ 1 + \frac{(y - 1)(y + 1)}{x^2} = 0 \]
\[ 1 + \frac{y^2 - 1}{x^2} = 0 \]
\[ \frac{y^2 - 1 + x^2}{x^2} = 0 \]
This leads to:
\[ y^2 + x^2 = 1 \]

8. **Conclusion**:
The equation \( x^2 + y^2 = 1 \) represents a circle with radius 1 centered at the origin in the complex plane. Therefore, the curve represented by the given condition is a circle.

### Final Answer:
The curve represented by the condition \( \arg\left(\frac{z - i}{z + i}\right) = \frac{\pi}{2} \) is a circle given by the equation:
\[ x^2 + y^2 = 1 \]