If the locus of the complex number `z` given by `arg(z+i)-arg(z-i)=(2pi)/(3)` is an arc of a circle, then the length of the arc is

Options

A. `(4pi)/(3)`
B. `(4pi)/(3sqrt3)`
C. `(2sqrt3)/(3)`
D. `(2pi)/(3sqrt3)`

Answer

To solve the problem, we need to analyze the given condition involving the complex number \( z \). The condition states:

\[ \arg(z + i) - \arg(z - i) = \frac{2\pi}{3} \]

### Step 1: Rewrite the Argument Condition
We can rewrite the argument condition using the properties of complex numbers. The expression can be interpreted as the argument of the quotient of two complex numbers:

\[ \arg\left(\frac{z + i}{z - i}\right) = \frac{2\pi}{3} \]

### Step 2: Express \( z \) in Cartesian Form
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then we can rewrite \( z + i \) and \( z - i \) as:

\[ z + i = x + i(y + 1) \]
\[ z - i = x + i(y - 1) \]

### Step 3: Calculate the Quotient
Now we need to compute the quotient:

\[ \frac{z + i}{z - i} = \frac{x + i(y + 1)}{x + i(y - 1)} \]

To simplify this, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{(x + i(y + 1))(x - i(y - 1))}{(x + i(y - 1))(x - i(y - 1))} \]

Calculating the denominator:

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

Calculating the numerator:

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

Combining the terms gives us:

\[ = x^2 + (y^2 - 1) + i(2xy) \]

Thus, we have:

\[ \frac{(x^2 + (y^2 - 1)) + i(2xy)}{x^2 + (y - 1)^2} \]

### Step 4: Find the Argument
The argument of a complex number \( a + bi \) is given by \( \tan^{-1}\left(\frac{b}{a}\right) \). Therefore, we need to find:

\[ \arg\left(\frac{(x^2 + (y^2 - 1)) + i(2xy)}{x^2 + (y - 1)^2}\right) = \tan^{-1}\left(\frac{2xy}{x^2 + (y^2 - 1)}\right) \]

### Step 5: Set the Argument Equal to \( \frac{2\pi}{3} \)
Setting the argument equal to \( \frac{2\pi}{3} \):

\[ \tan^{-1}\left(\frac{2xy}{x^2 + (y^2 - 1)}\right) = \frac{2\pi}{3} \]

This implies that:

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

### Step 6: Solve for the Locus
From this equation, we can derive the locus of points \( z \) that satisfy this condition. The locus will be a circular arc.

### Step 7: Determine the Length of the Arc
The angle \( \frac{2\pi}{3} \) corresponds to a circular arc. If we denote the radius of the circle as \( r \), the length \( L \) of the arc can be calculated using the formula:

\[ L = r \cdot \theta \]

where \( \theta \) is in radians. Here, \( \theta = \frac{2\pi}{3} \).

### Conclusion
To find the exact length of the arc, we need the radius \( r \). Assuming \( r = 1 \) (for simplicity), the length of the arc would be:

\[ L = 1 \cdot \frac{2\pi}{3} = \frac{2\pi}{3} \]