The radius of the circle `|(z-i)/(z+i)|=5` is given by

Options

A. `(13)/(12)`
B. `(5)/(12)`
C. 5
D. 625

Answer

To find the radius of the circle defined by the equation \(|(z-i)/(z+i)|=5\), we can follow these steps:

### Step 1: Rewrite the complex number
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can rewrite the given equation:
\[ \left| \frac{(x + iy) - i}{(x + iy) + i} \right| = 5 \]
This simplifies to:
\[ \left| \frac{x + (y - 1)i}{x + (y + 1)i} \right| = 5 \]

### Step 2: Use the property of modulus
Using the property of modulus of complex numbers, we can express the equation as:
\[ \frac{\sqrt{x^2 + (y - 1)^2}}{\sqrt{x^2 + (y + 1)^2}} = 5 \]

### Step 3: Square both sides
Squaring both sides to eliminate the square roots gives:
\[ \frac{x^2 + (y - 1)^2}{x^2 + (y + 1)^2} = 25 \]

### Step 4: Cross-multiply
Cross-multiplying results in:
\[ x^2 + (y - 1)^2 = 25(x^2 + (y + 1)^2) \]

### Step 5: Expand both sides
Expanding both sides, we have:
\[ x^2 + (y^2 - 2y + 1) = 25(x^2 + y^2 + 2y + 1) \]
This simplifies to:
\[ x^2 + y^2 - 2y + 1 = 25x^2 + 25y^2 + 50y + 25 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ x^2 + y^2 - 2y + 1 - 25x^2 - 25y^2 - 50y - 25 = 0 \]
Combining like terms results in:
\[ -24x^2 - 24y^2 - 48y - 24 = 0 \]

### Step 7: Divide by -24
Dividing the entire equation by -24 simplifies to:
\[ x^2 + y^2 + 2y + 1 = 0 \]

### Step 8: Complete the square
To complete the square for the \(y\) terms:
\[ x^2 + (y + 1)^2 = 0 \]

### Step 9: Identify the center and radius
This represents a circle centered at \((0, -1)\) with radius \(0\). However, we need to find the radius in terms of the original equation.

### Step 10: Determine the radius from the original equation
From the original equation \(|(z-i)/(z+i)|=5\), we can see that the radius is \(5\).

### Final Answer
Thus, the radius of the circle is \(5\).

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