Let a circle C in complex plane pass through the points `z_1=3+4i, z_2=4+3i` and `z_3=5i`. If `z(nez_1)` is a point on C such that the line through `z and z_1` is perpendicular to the line through `z_2 and z_3` , then arg(z) is equal to :

Options

A. `tan^(-1)(2/sqrt5)-pi`
B. `tan^(-1)(24/7)-pi`
C. `tan^(-1)(3)-pi`
D. `tan^(-1)(3/4)-pi`

Answer

To solve the problem, we need to find the argument of the point \( z \) on the circle \( C \) that passes through the points \( z_1 = 3 + 4i \), \( z_2 = 4 + 3i \), and \( z_3 = 5i \). The line through \( z \) and \( z_1 \) is perpendicular to the line through \( z_2 \) and \( z_3 \).

### Step 1: Find the slope of the line through \( z_2 \) and \( z_3 \)

The points are:
- \( z_2 = 4 + 3i \) (which corresponds to the point \( (4, 3) \))
- \( z_3 = 5i \) (which corresponds to the point \( (0, 5) \))

The slope \( m \) of the line through \( z_2 \) and \( z_3 \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 5}{4 - 0} = \frac{-2}{4} = -\frac{1}{2} \]

### Step 2: Find the slope of the line through \( z \) and \( z_1 \)

Let \( z = x + yi \). The slope of the line through \( z \) and \( z_1 \) is:
\[ m_{z z_1} = \frac{y - 4}{x - 3} \]

### Step 3: Use the perpendicularity condition

Since the lines are perpendicular, we have:
\[ m_{z z_1} \cdot m_{z_2 z_3} = -1 \]
Substituting the slopes:
\[ \frac{y - 4}{x - 3} \cdot \left(-\frac{1}{2}\right) = -1 \]
This simplifies to:
\[ \frac{y - 4}{x - 3} = 2 \quad \text{(1)} \]

### Step 4: Express \( y \) in terms of \( x \)

From equation (1), we can express \( y \):
\[ y - 4 = 2(x - 3) \implies y = 2x - 6 + 4 = 2x - 2 \]

### Step 5: Find the equation of the circle

The circle passes through the points \( z_1, z_2, z_3 \). The center of the circle is at the origin (0, 0) and the radius can be calculated using the distance from the origin to any of the points. The distance from the origin to \( z_1 \) is:
\[ r = |z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \]
Thus, the equation of the circle is:
\[ x^2 + y^2 = 25 \quad \text{(2)} \]

### Step 6: Substitute \( y \) into the circle's equation

Substituting \( y = 2x - 2 \) into equation (2):
\[ x^2 + (2x - 2)^2 = 25 \]
Expanding this:
\[ x^2 + (4x^2 - 8x + 4) = 25 \]
Combining like terms:
\[ 5x^2 - 8x + 4 - 25 = 0 \implies 5x^2 - 8x - 21 = 0 \]

### Step 7: Solve the quadratic equation

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 5 \cdot (-21)}}{2 \cdot 5} = \frac{8 \pm \sqrt{64 + 420}}{10} = \frac{8 \pm \sqrt{484}}{10} = \frac{8 \pm 22}{10} \]
Calculating the two possible values for \( x \):
1. \( x = \frac{30}{10} = 3 \)
2. \( x = \frac{-14}{10} = -1.4 \)

### Step 8: Find corresponding \( y \) values

For \( x = 3 \):
\[ y = 2(3) - 2 = 4 \]
For \( x = -1.4 \):
\[ y = 2(-1.4) - 2 = -4.8 \]

### Step 9: Find the arguments of \( z \)

1. For \( z = 3 + 4i \):
\[ \arg(z) = \tan^{-1}\left(\frac{4}{3}\right) \]
2. For \( z = -1.4 - 4.8i \):
\[ \arg(z) = \tan^{-1}\left(\frac{-4.8}{-1.4}\right) + \pi = \tan^{-1}\left(\frac{24}{7}\right) + \pi \]

### Final Step: Determine the correct argument

Since \( z = -1.4 - 4.8i \) lies in the third quadrant, we need to adjust the argument:
\[ \arg(z) = \tan^{-1}\left(\frac{24}{7}\right) - \pi \]

Thus, the final answer is:
\[ \arg(z) = \tan^{-1}\left(\frac{24}{7}\right) - \pi \]