If` |(z +4)/(2z -1)|=1,` where `z =x +iy.` Then the point (x,y) lies on a:

Options

A. circle with center (4,0)
B. circle with center (-2,0)
C. circle with center (2,0)
D. None of these

Explanation

To solve the equation \( \left| \frac{z + 4}{2z - 1} \right| = 1 \) where \( z = x + iy \), we can follow these steps:

### Step 1: Rewrite the equation using \( z = x + iy \)
We start by substituting \( z \) into the equation:
\[ \left| \frac{(x + iy) + 4}{2(x + iy) - 1} \right| = 1 \]
This simplifies to:
\[ \left| \frac{(x + 4) + iy}{(2x - 1) + 2iy} \right| = 1 \]

### Step 2: Use the property of modulus
The property of modulus states that \( |A| = |B| \) implies \( |A|^2 = |B|^2 \). Therefore, we can square both sides:
\[ \left| (x + 4) + iy \right|^2 = \left| (2x - 1) + 2iy \right|^2 \]

### Step 3: Calculate the squares of the moduli
Calculating the left-hand side:
\[ (x + 4)^2 + y^2 \]
Calculating the right-hand side:
\[ (2x - 1)^2 + (2y)^2 = (2x - 1)^2 + 4y^2 \]

### Step 4: Set the equations equal
Now we set the two expressions equal to each other:
\[ (x + 4)^2 + y^2 = (2x - 1)^2 + 4y^2 \]

### Step 5: Expand both sides
Expanding both sides:
\[ (x^2 + 8x + 16 + y^2) = (4x^2 - 4x + 1 + 4y^2) \]

### Step 6: Rearranging the equation
Rearranging gives:
\[ x^2 + 8x + 16 + y^2 - 4x^2 + 4x - 1 - 4y^2 = 0 \]
Combining like terms:
\[ -3x^2 - 3y^2 + 12x + 15 = 0 \]

### Step 7: Simplifying the equation
Dividing the entire equation by -3:
\[ x^2 + y^2 - 4x - 5 = 0 \]

### Step 8: Completing the square
To complete the square for \( x \):
\[ (x^2 - 4x) + y^2 = 5 \]
Completing the square for \( x \):
\[ (x - 2)^2 - 4 + y^2 = 5 \]
Thus:
\[ (x - 2)^2 + y^2 = 9 \]

### Conclusion
This represents a circle with center \( (2, 0) \) and radius \( 3 \).

### Final Answer
The point \( (x, y) \) lies on a circle with center \( (2, 0) \) and radius \( 3 \).

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