Find the locus of a complex number z such that arg `((z-2)/(z+2))= (pi)/(3)`

Answer

To find the locus of the complex number \( z \) such that \( \arg\left(\frac{z-2}{z+2}\right) = \frac{\pi}{3} \), we can follow these steps:

### Step 1: Set up the equation
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. The equation becomes:
\[ \arg\left(\frac{(x + iy) - 2}{(x + iy) + 2}\right) = \frac{\pi}{3} \]

### Step 2: Simplify the expression
This can be rewritten as:
\[ \arg\left(\frac{(x - 2) + iy}{(x + 2) + iy}\right) = \frac{\pi}{3} \]

### Step 3: Use the property of arguments
The argument of a complex number \( \frac{a + bi}{c + di} \) can be expressed as:
\[ \arg(a + bi) - \arg(c + di) \]
Thus, we have:
\[ \arg((x - 2) + iy) - \arg((x + 2) + iy) = \frac{\pi}{3} \]

### Step 4: Express the arguments
Let:
\[ \theta_1 = \arg((x - 2) + iy) \]
\[ \theta_2 = \arg((x + 2) + iy) \]
Then:
\[ \theta_1 - \theta_2 = \frac{\pi}{3} \]

### Step 5: Use the tangent function
We can express the arguments in terms of tangent:
\[ \tan(\theta_1) = \frac{y}{x - 2}, \quad \tan(\theta_2) = \frac{y}{x + 2} \]
Using the tangent subtraction formula:
\[ \tan(\theta_1 - \theta_2) = \frac{\tan(\theta_1) - \tan(\theta_2}{1 + \tan(\theta_1)\tan(\theta_2)} = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Step 6: Substitute the values
Substituting the values of \( \tan(\theta_1) \) and \( \tan(\theta_2) \):
\[ \frac{\frac{y}{x - 2} - \frac{y}{x + 2}}{1 + \frac{y}{x - 2} \cdot \frac{y}{x + 2}} = \sqrt{3} \]

### Step 7: Simplify the equation
This simplifies to:
\[ \frac{y\left(\frac{(x + 2) - (x - 2)}{(x - 2)(x + 2)}\right)}{1 + \frac{y^2}{(x - 2)(x + 2)}} = \sqrt{3} \]
\[ \frac{y\left(\frac{4}{(x - 2)(x + 2)}\right)}{1 + \frac{y^2}{(x - 2)(x + 2)}} = \sqrt{3} \]

### Step 8: Cross-multiply and simplify
Cross-multiplying gives:
\[ 4y = \sqrt{3}\left((x - 2)(x + 2) + y^2\right) \]
Expanding and rearranging leads to:
\[ \sqrt{3}x^2 + \sqrt{3}y^2 - 4y - 4\sqrt{3} = 0 \]

### Final Step: Identify the locus
This is the equation of a conic section. Rearranging gives the locus of the complex number \( z \):
\[ \sqrt{3}x^2 + \sqrt{3}y^2 - 4y - 4\sqrt{3} = 0 \]