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
Start with the given equation:
\[ \left| \frac{z - i}{z + i} \right| = 3 \]

### Step 2: Apply the modulus property
Using the property of modulus, we can rewrite this as:
\[ |z - i| = 3 |z + i| \]

### Step 3: Cross-multiply
Cross-multiplying gives:
\[ |z - i| = 3 |z + i| \]

### Step 4: Square both sides
To eliminate the modulus, square both sides:
\[ |z - i|^2 = 9 |z + i|^2 \]

### Step 5: Express in terms of \( z \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then:
\[ |z - i|^2 = |(x + i(y - 1))|^2 = x^2 + (y - 1)^2 \]
\[ |z + i|^2 = |(x + i(y + 1))|^2 = x^2 + (y + 1)^2 \]

### Step 6: Substitute back into the equation
Substituting these into the equation gives:
\[ x^2 + (y - 1)^2 = 9(x^2 + (y + 1)^2) \]

### Step 7: Expand both sides
Expanding both sides:
\[ x^2 + (y^2 - 2y + 1) = 9(x^2 + (y^2 + 2y + 1)) \]
\[ x^2 + y^2 - 2y + 1 = 9x^2 + 9y^2 + 18y + 9 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ x^2 + y^2 - 2y + 1 - 9x^2 - 9y^2 - 18y - 9 = 0 \]
\[ -8x^2 - 8y^2 - 16y - 8 = 0 \]

### Step 9: Simplify the equation
Dividing through by -8:
\[ x^2 + y^2 + 2y + 1 = 0 \]

### Step 10: Complete the square
Completing the square for the \( y \) terms:
\[ x^2 + (y + 1)^2 = 0 \]

### Step 11: Identify the center and radius
This represents a circle centered at \( (0, -1) \) with a radius of \( 0 \). However, we need to find the radius from the original equation.

### Step 12: Radius from the original equation
From the original equation \( |z - i| = 3 |z + i| \), we can see that the radius of the circle is \( 3 \).

Thus, the radius of the circle is:
\[ \text{Radius} = 3 \]