If `|(z+i)/(z-i)|=sqrt(3)`, then z lies on a circle whose radius, is

Options

A. `2/sqrt(21)`
B. `1/sqrt(21)`
C. `sqrt(3)`
D. `sqrt(21)`

Explanation

To solve the problem \( \left| \frac{z+i}{z-i} \right| = \sqrt{3} \), we will express \( z \) as a complex number and manipulate the equation step by step.

### Step 1: Express \( z \) in terms of real and imaginary parts
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Substitute \( z \) into the equation
Substituting \( z \) into the equation gives:
\[ \left| \frac{(x + iy) + i}{(x + iy) - i} \right| = \sqrt{3} \]
This simplifies to:
\[ \left| \frac{x + i(y + 1)}{x + i(y - 1)} \right| = \sqrt{3} \]

### Step 3: Calculate the modulus of the fraction
The modulus of a complex fraction \( \frac{a + bi}{c + di} \) is given by:
\[ \left| \frac{a + bi}{c + di} \right| = \frac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}} \]
Thus, we have:
\[ \left| \frac{x + i(y + 1)}{x + i(y - 1)} \right| = \frac{\sqrt{x^2 + (y + 1)^2}}{\sqrt{x^2 + (y - 1)^2}} = \sqrt{3} \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \frac{x^2 + (y + 1)^2}{x^2 + (y - 1)^2} = 3 \]

### Step 5: Cross-multiply to simplify
Cross-multiplying results in:
\[ x^2 + (y + 1)^2 = 3(x^2 + (y - 1)^2) \]

### Step 6: Expand both sides
Expanding both sides:
\[ x^2 + (y^2 + 2y + 1) = 3(x^2 + (y^2 - 2y + 1)) \]
This simplifies to:
\[ x^2 + y^2 + 2y + 1 = 3x^2 + 3y^2 - 6y + 3 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ 0 = 3x^2 + 3y^2 - 6y + 3 - x^2 - y^2 - 2y - 1 \]
Which simplifies to:
\[ 0 = 2x^2 + 2y^2 - 8y + 2 \]

### Step 8: Divide by 2
Dividing the entire equation by 2 results in:
\[ 0 = x^2 + y^2 - 4y + 1 \]

### Step 9: Complete the square for \( y \)
Completing the square for \( y \):
\[ 0 = x^2 + (y^2 - 4y + 4) - 4 + 1 \]
This simplifies to:
\[ 0 = x^2 + (y - 2)^2 - 3 \]
Thus, we have:
\[ x^2 + (y - 2)^2 = 3 \]

### Step 10: Identify the center and radius of the circle
This equation represents a circle with center \( (0, 2) \) and radius \( \sqrt{3} \).

### Final Answer
The radius of the circle is \( \sqrt{3} \).

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