In the Argand plane `|(z-i)/(z+i)|` = 4 represents a

Options

A. pair of distinct lines
B. circle
C. a pair of coincident
D. none of these

Answer

To solve the problem \( \left| \frac{z - i}{z + i} \right| = 4 \) in the Argand plane, we will proceed step by step.

### Step 1: Rewrite the expression in terms of \( z \)
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can rewrite the expression:
\[ \left| \frac{(x + iy) - i}{(x + iy) + i} \right| = \left| \frac{x + (y - 1)i}{x + (y + 1)i} \right| \]

### Step 2: Use the property of moduli
Using the property of moduli, we know that:
\[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \]
Thus, we have:
\[ \frac{|x + (y - 1)i|}{|x + (y + 1)i|} = 4 \]

### Step 3: Calculate the moduli
Now, we calculate the moduli:
\[ |x + (y - 1)i| = \sqrt{x^2 + (y - 1)^2} \]
\[ |x + (y + 1)i| = \sqrt{x^2 + (y + 1)^2} \]
So, we can rewrite the equation as:
\[ \frac{\sqrt{x^2 + (y - 1)^2}}{\sqrt{x^2 + (y + 1)^2}} = 4 \]

### Step 4: Square both sides
Squaring both sides gives:
\[ \frac{x^2 + (y - 1)^2}{x^2 + (y + 1)^2} = 16 \]

### Step 5: Cross-multiply
Cross-multiplying leads to:
\[ x^2 + (y - 1)^2 = 16(x^2 + (y + 1)^2) \]

### Step 6: Expand both sides
Expanding both sides:
\[ x^2 + (y^2 - 2y + 1) = 16(x^2 + (y^2 + 2y + 1)) \]
This simplifies to:
\[ x^2 + y^2 - 2y + 1 = 16x^2 + 16y^2 + 32y + 16 \]

### Step 7: Rearranging the equation
Rearranging gives:
\[ x^2 - 16x^2 + y^2 - 16y^2 - 2y - 32y + 1 - 16 = 0 \]
\[ -15x^2 - 15y^2 - 34y - 15 = 0 \]

### Step 8: Divide by -15
Dividing the entire equation by -15 results in:
\[ x^2 + y^2 + \frac{34}{15}y + 1 = 0 \]

### Step 9: Completing the square
To complete the square for \( y \):
\[ x^2 + \left(y + \frac{17}{15}\right)^2 - \left(\frac{17}{15}\right)^2 + 1 = 0 \]
This leads to:
\[ x^2 + \left(y + \frac{17}{15}\right)^2 = \left(\frac{17}{15}\right)^2 - 1 \]

### Step 10: Final form
This represents a circle in the form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \( h = 0 \), \( k = -\frac{17}{15} \), and \( r^2 = \left(\frac{17}{15}\right)^2 - 1 \).

### Conclusion
Thus, the equation \( \left| \frac{z - i}{z + i} \right| = 4 \) represents a circle in the Argand plane.

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