If ` z_(1),z_(2),z_(3),z_(4)` are two pairs of conjugate complex numbers, then `arg(z_(1)/z_(3)) + arg(z_(2)/z_(4))` is

Options

A. 0
B. `pi/2`
C. `(3pi)/4`
D. `pi`

Answer

To solve the problem, we need to find the value of the expression \( \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) \) where \( z_1, z_2 \) are conjugate complex numbers and \( z_3, z_4 \) are also conjugate complex numbers.

### Step 1: Define the complex numbers
Let:
- \( z_1 = a + ib \)
- \( z_2 = a - ib \) (conjugate of \( z_1 \))
- \( z_3 = x + iy \)
- \( z_4 = x - iy \) (conjugate of \( z_3 \))

### Step 2: Compute \( \frac{z_1}{z_3} \)
To find \( \frac{z_1}{z_3} \):
\[ \frac{z_1}{z_3} = \frac{a + ib}{x + iy} \]
We rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:
\[ \frac{z_1}{z_3} = \frac{(a + ib)(x - iy)}{(x + iy)(x - iy)} = \frac{(ax + by) + i(bx - ay)}{x^2 + y^2} \]

### Step 3: Find \( \arg\left(\frac{z_1}{z_3}\right) \)
The argument of a complex number \( \frac{u + iv}{r} \) is given by \( \tan^{-1}\left(\frac{v}{u}\right) \):
\[ \arg\left(\frac{z_1}{z_3}\right) = \tan^{-1}\left(\frac{bx - ay}{ax + by}\right) \]

### Step 4: Compute \( \frac{z_2}{z_4} \)
Now, compute \( \frac{z_2}{z_4} \):
\[ \frac{z_2}{z_4} = \frac{a - ib}{x - iy} \]
Again, we rationalize the denominator:
\[ \frac{z_2}{z_4} = \frac{(a - ib)(x + iy)}{(x - iy)(x + iy)} = \frac{(ax + by) - i(bx - ay)}{x^2 + y^2} \]

### Step 5: Find \( \arg\left(\frac{z_2}{z_4}\right) \)
The argument is:
\[ \arg\left(\frac{z_2}{z_4}\right) = \tan^{-1}\left(\frac{ay - bx}{ax + by}\right) \]

### Step 6: Combine the arguments
Now, we need to add the two arguments:
\[ \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) = \tan^{-1}\left(\frac{bx - ay}{ax + by}\right) + \tan^{-1}\left(\frac{ay - bx}{ax + by}\right) \]

### Step 7: Use the tangent addition formula
Using the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Let \( A = \arg\left(\frac{z_1}{z_3}\right) \) and \( B = \arg\left(\frac{z_2}{z_4}\right) \):
\[ \tan(A + B) = \frac{\frac{bx - ay}{ax + by} + \frac{ay - bx}{ax + by}}{1 - \left(\frac{bx - ay}{ax + by}\right)\left(\frac{ay - bx}{ax + by}\right)} \]
The numerator simplifies to:
\[ \frac{(bx - ay) + (ay - bx)}{ax + by} = 0 \]
Thus, \( \tan(A + B) = 0 \), which implies:
\[ A + B = 0 \quad \Rightarrow \quad \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) = 0 \]

### Final Result
Therefore, the value of \( \arg\left(\frac{z_1}{z_3}\right) + \arg\left(\frac{z_2}{z_4}\right) \) is:
\[ \boxed{0} \]