The locus of the points z satisfying the condition arg `((z-1)/(z+1))=pi/3` is, a

Options

A. A straight line
B. Circle
C. A parabola
D. Ellipse

Answer

To find the locus of the points \( z \) satisfying the condition \( \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{3} \), we will follow these steps:

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

### Step 2: Calculate \( z - 1 \) and \( z + 1 \)
\[ z - 1 = (x - 1) + iy \]
\[ z + 1 = (x + 1) + iy \]

### Step 3: Write the expression for \( \frac{z - 1}{z + 1} \)
\[ \frac{z - 1}{z + 1} = \frac{(x - 1) + iy}{(x + 1) + iy} \]

### Step 4: Rationalize the denominator
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{((x - 1) + iy)((x + 1) - iy)}{((x + 1) + iy)((x + 1) - iy)} \]

### Step 5: Simplify the denominator
The denominator becomes:
\[ (x + 1)^2 + y^2 \]

### Step 6: Simplify the numerator
The numerator simplifies to:
\[ (x^2 - 1) + i(2y) \]
Thus, we have:
\[ \frac{(x^2 - 1) + i(2y)}{(x + 1)^2 + y^2} \]

### Step 7: Identify the real and imaginary parts
The real part \( R \) and imaginary part \( I \) are:
\[ R = \frac{x^2 - 1}{(x + 1)^2 + y^2}, \quad I = \frac{2y}{(x + 1)^2 + y^2} \]

### Step 8: Use the argument condition
From the condition \( \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{3} \), we know:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]
Thus,
\[ \frac{I}{R} = \sqrt{3} \]
Substituting the values of \( I \) and \( R \):
\[ \frac{\frac{2y}{(x + 1)^2 + y^2}}{\frac{x^2 - 1}{(x + 1)^2 + y^2}} = \sqrt{3} \]
This simplifies to:
\[ \frac{2y}{x^2 - 1} = \sqrt{3} \]

### Step 9: Cross-multiply and rearrange
Cross-multiplying gives:
\[ 2y = \sqrt{3}(x^2 - 1) \]
Rearranging this yields:
\[ \sqrt{3}x^2 - 2y - \sqrt{3} = 0 \]

### Step 10: Complete the square
Rearranging the equation:
\[ \sqrt{3}x^2 - 2y = \sqrt{3} \]
This can be rewritten as:
\[ \sqrt{3}x^2 = 2y + \sqrt{3} \]
Dividing through by \( \sqrt{3} \):
\[ x^2 = \frac{2}{\sqrt{3}}y + 1 \]

### Step 11: Identify the locus
This equation represents a parabola opening upwards.

### Conclusion
The locus of the points \( z \) satisfying the condition \( \arg\left(\frac{z-1}{z+1}\right) = \frac{\pi}{3} \) is a parabola.