Find the locus of z satisfying `|(z-3)/(z+1)|=3` in the complex plane.

Answer

To find the locus of the complex number \( z \) satisfying the equation

\[ \left| \frac{z-3}{z+1} \right| = 3 \]

we will follow these steps:

### Step 1: Substitute \( z \) with \( x + iy \)

Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we can rewrite the equation as:

\[ \left| \frac{(x + iy) - 3}{(x + iy) + 1} \right| = 3 \]

### Step 2: Simplify the expression

This simplifies to:

\[ \left| \frac{(x - 3) + iy}{(x + 1) + iy} \right| = 3 \]

### Step 3: Apply the modulus property

Using the property of modulus of complex numbers, we can express this as:

\[ \frac{\sqrt{(x - 3)^2 + y^2}}{\sqrt{(x + 1)^2 + y^2}} = 3 \]

### Step 4: Square both sides

Squaring both sides gives:

\[ \frac{(x - 3)^2 + y^2}{(x + 1)^2 + y^2} = 9 \]

### Step 5: Cross-multiply

Cross-multiplying leads to:

\[ (x - 3)^2 + y^2 = 9 \left( (x + 1)^2 + y^2 \right) \]

### Step 6: Expand both sides

Expanding both sides results in:

\[ (x - 3)^2 + y^2 = 9(x^2 + 2x + 1 + y^2) \]

This simplifies to:

\[ x^2 - 6x + 9 + y^2 = 9x^2 + 18x + 9 + 9y^2 \]

### Step 7: Rearrange the equation

Rearranging gives:

\[ x^2 - 9x^2 - 6x - 18x + 9 + 9 - 9y^2 + y^2 = 0 \]

This simplifies to:

\[ -8x^2 - 24x - 8y^2 + 0 = 0 \]

### Step 8: Divide through by -8

Dividing through by -8 leads to:

\[ x^2 + 3x + y^2 = 0 \]

### Step 9: Complete the square

Completing the square for \( x \):

\[ (x + \frac{3}{2})^2 - \frac{9}{4} + y^2 = 0 \]

This can be rearranged to:

\[ (x + \frac{3}{2})^2 + y^2 = \frac{9}{4} \]

### Step 10: Interpret the result

This is the equation of a circle with center at \( (-\frac{3}{2}, 0) \) and radius \( \frac{3}{2} \).

### Final Answer

The locus of \( z \) satisfying \( \left| \frac{z-3}{z+1} \right| = 3 \) is a circle centered at \( (-\frac{3}{2}, 0) \) with radius \( \frac{3}{2} \).

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