Key Concepts and Formulas

• Circle in the Complex Plane: If $|z - c| = r$, then the complex number $z$ lies on a circle with center $c$ and radius $r$.
• Perpendicular Lines: Two lines with slopes $m_1$ and $m_2$ are perpendicular if and only if $m_1 m_2 = -1$. In the complex plane, the slopes of the lines joining $z_1$ and $z$ and $z_2$ and $z_3$ are perpendicular if $\frac{z - z_1}{\overline{z} - \overline{z_1}} / \frac{z_3 - z_2}{\overline{z_3} - \overline{z_2}}$ is purely imaginary. Alternatively, $arg(\frac{z-z_1}{z_3-z_2}) = \pm \frac{\pi}{2}$.
• Argument of a Complex Number: For a complex number $z = x + iy$, its principal argument, denoted $\operatorname{Arg}(z)$, is the unique angle $\theta$ in the interval $(-\pi, \pi]$ such that $x = |z|\cos\theta$ and $y = |z|\sin\theta$. We can use $\operatorname{Arg}(z) = \tan^{-1}(y/x)$, remembering to adjust the angle based on the quadrant.

Step-by-Step Solution

1. Finding the Equation of the Circle C

We are given three points on the circle: $z_1 = 3 + 4i$, $z_2 = 4 + 3i$, and $z_3 = 5i$. Let the equation of the circle be $|z - c| = r$, where $c$ is the center and $r$ is the radius. Since all three points lie on the circle, we have: $|z_1 - c| = |z_2 - c| = |z_3 - c| = r$

Why this step is taken: To define the circle, we need to find its center and radius. Using the given points that lie on the circle allows us to formulate equations in terms of the unknown center and radius.

We have $|z_1| = \sqrt{3^2 + 4^2} = 5$, $|z_2| = \sqrt{4^2 + 3^2} = 5$, and $|z_3| = \sqrt{0^2 + 5^2} = 5$. Thus, the circle passes through the origin and is centered at the origin, meaning $c = 0$ and $r = 5$. The equation of the circle is $|z| = 5$, or $x^2 + y^2 = 25$.

2. Using the Perpendicularity Condition

The line through $z$ and $z_1$ is perpendicular to the line through $z_2$ and $z_3$. This means $arg(\frac{z - z_1}{z_3 - z_2}) = \pm \frac{\pi}{2}$. So, $\frac{z - z_1}{z_3 - z_2}$ is purely imaginary.

Why this step is taken: This step translates the geometric condition (perpendicular lines) into an algebraic condition involving complex numbers. This is a key step in solving the problem.

We have $z_3 - z_2 = 5i - (4 + 3i) = -4 + 2i$. Let $z = x + iy$. Then $z - z_1 = (x + iy) - (3 + 4i) = (x - 3) + i(y - 4)$. Then $\frac{z - z_1}{z_3 - z_2} = \frac{(x - 3) + i(y - 4)}{-4 + 2i}$. Multiplying the numerator and denominator by the conjugate of the denominator, we get: $\frac{((x - 3) + i(y - 4))(-4 - 2i)}{(-4 + 2i)(-4 - 2i)} = \frac{-4(x - 3) + 2(y - 4) + i(-2(x - 3) - 4(y - 4))}{16 + 4}$ $= \frac{-4x + 12 + 2y - 8 + i(-2x + 6 - 4y + 16)}{20} = \frac{-4x + 2y + 4 + i(-2x - 4y + 22)}{20}$ Since this is purely imaginary, the real part must be zero: $-4x + 2y + 4 = 0$, which simplifies to $2x - y - 2 = 0$, or $y = 2x - 2$.

3. Finding the Point $z$ as the Intersection of the Circle and the Line

The point $z$ must satisfy two conditions:

1. It lies on the circle C: $x^2 + y^2 = 25$
2. It lies on the line L: $y = 2x - 2$

Why this step is taken: The point $z$ is defined by both its presence on the circle C and by the perpendicularity condition which led to line L. Thus, $z$ must be an intersection point of the circle and the line.

Substitute $y = 2x - 2$ into $x^2 + y^2 = 25$: $x^2 + (2x - 2)^2 = 25$ $x^2 + 4x^2 - 8x + 4 = 25$ $5x^2 - 8x - 21 = 0$ $(5x + 7)(x - 3) = 0$ So $x = 3$ or $x = -\frac{7}{5}$.

If $x = 3$, then $y = 2(3) - 2 = 4$, so $z = 3 + 4i = z_1$. If $x = -\frac{7}{5}$, then $y = 2(-\frac{7}{5}) - 2 = -\frac{14}{5} - \frac{10}{5} = -\frac{24}{5}$. So $z = -\frac{7}{5} - \frac{24}{5}i$.

Since $z \ne z_1$, we have $z = -\frac{7}{5} - \frac{24}{5}i$.

**4. Calculating the Argument of $z$}

We have $z = -\frac{7}{5} - i\frac{24}{5}$. Both the real and imaginary parts are negative, so $z$ lies in the third quadrant.

Why this step is taken: The final step requires finding the argument of $z$. The quadrant of the point is crucial for determining the correct argument formula.

$\operatorname{Arg}(z) = \tan^{-1}(\frac{-24/5}{-7/5}) - \pi = \tan^{-1}(\frac{24}{7}) - \pi$.

Common Mistakes & Tips

• Quadrant Awareness: Always check the quadrant of the complex number when finding the argument. The arctangent function only returns values in the range $(-\pi/2, \pi/2)$, so you may need to add or subtract $\pi$ to get the correct argument.
• Checking for Simplifications: Look for ways to simplify the equation of the circle early on. Recognizing that all three given points have the same modulus simplifies the problem significantly.
• Using All Conditions: Remember to use all the given information, including the condition $z \neq z_1$, to select the correct solution.

Summary

We first found the equation of the circle by using the fact that the given points had the same modulus. Then, we used the perpendicularity condition to find a line that $z$ must lie on. Solving for the intersection of the line and the circle gave us two possible values for $z$, and we used the condition $z \ne z_1$ to choose the correct value. Finally, we calculated the argument of $z$, taking into account that it lies in the third quadrant. The final answer is $\tan^{-1}(\frac{24}{7}) - \pi$.

Final Answer

The final answer is \boxed{{\tan ^{ - 1}}\left( {{{24} \over 7}} \right) - \pi}, which corresponds to option (B).