The radius of the circle `|(z-i)/(z+i)|=3`, is

Options

A. `5/4`
B. `3/4`
C. `1/4`
D. none of these

Explanation

To find the radius of the circle defined by the equation \( \left| \frac{z - i}{z + i} \right| = 3 \), we can follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \left| \frac{z - i}{z + i} \right| = 3 \]
This can be rewritten as:
\[ \left| z - i \right| = 3 \left| z + i \right| \]

### Step 2: Let \( z = x + iy \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can express the magnitudes:
\[ \left| z - i \right| = \left| x + (y - 1)i \right| = \sqrt{x^2 + (y - 1)^2} \]
\[ \left| z + i \right| = \left| x + (y + 1)i \right| = \sqrt{x^2 + (y + 1)^2} \]

### Step 3: Substitute into the Equation
Substituting these into our equation gives:
\[ \sqrt{x^2 + (y - 1)^2} = 3 \sqrt{x^2 + (y + 1)^2} \]

### Step 4: Square Both Sides
Now, we square both sides to eliminate the square roots:
\[ x^2 + (y - 1)^2 = 9(x^2 + (y + 1)^2) \]

### Step 5: Expand Both Sides
Expanding both sides:
\[ x^2 + (y^2 - 2y + 1) = 9(x^2 + (y^2 + 2y + 1)) \]
This simplifies to:
\[ x^2 + y^2 - 2y + 1 = 9x^2 + 9y^2 + 18y + 9 \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ x^2 + y^2 - 2y + 1 - 9x^2 - 9y^2 - 18y - 9 = 0 \]
Combining like terms:
\[ -8x^2 - 8y^2 + 16y - 8 = 0 \]

### Step 7: Divide by -8
Dividing the entire equation by -8:
\[ x^2 + y^2 - 2y + 1 = 0 \]

### Step 8: Complete the Square
To complete the square for the \( y \) terms:
\[ x^2 + (y^2 - 2y + 1) = 0 \]
This can be rewritten as:
\[ x^2 + (y - 1)^2 = 0 \]

### Step 9: Identify the Circle
The equation \( x^2 + (y - 1)^2 = 0 \) represents a circle with center at \( (0, 1) \) and radius \( 0 \).

### Step 10: Conclusion
Thus, the radius of the circle is:
\[ \text{Radius} = 0 \]