Key Concepts and Formulas

• Complex Number Representation: A complex number $z$ can be represented as $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part.
• Argument of a Complex Number: For a complex number $z = x + iy$, the argument $\arg(z) = \theta$ is the angle that the vector representing $z$ makes with the positive real axis. $\tan(\theta) = \frac{y}{x}$.
• Standard Equation of a Circle: The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$.

Step-by-Step Solution

• Step 1: Substitute $z = x + iy$ We express the complex number $z$ in terms of its real and imaginary parts. This allows us to convert the complex equation into a Cartesian equation. $z = x + iy$ Substitute this into the given expression: $\frac{z - 1}{z + 1} = \frac{(x + iy) - 1}{(x + iy) + 1} = \frac{(x - 1) + iy}{(x + 1) + iy}$

• Step 2: Simplify the Complex Fraction To find the argument, we need the complex number in the form $A + iB$. We multiply the numerator and denominator by the conjugate of the denominator to rationalize the denominator. The conjugate of $(x + 1) + iy$ is $(x + 1) - iy$. $\frac{(x - 1) + iy}{(x + 1) + iy} \times \frac{(x + 1) - iy}{(x + 1) - iy}$ Expanding the numerator: $((x - 1) + iy)((x + 1) - iy) = (x - 1)(x + 1) - iy(x - 1) + iy(x + 1) - i^2y^2$ $= x^2 - 1 - ixy + iy + ixy + iy + y^2 = x^2 + y^2 - 1 + 2iy$ Expanding the denominator: $((x + 1) + iy)((x + 1) - iy) = (x + 1)^2 - (iy)^2 = (x + 1)^2 + y^2$ Therefore, $\frac{z - 1}{z + 1} = \frac{x^2 + y^2 - 1}{(x + 1)^2 + y^2} + i \frac{2y}{(x + 1)^2 + y^2}$

• Step 3: Apply the Argument Condition We are given $\arg\left(\frac{z - 1}{z + 1}\right) = \frac{\pi}{4}$. This means that the angle of the complex number $\frac{z - 1}{z + 1}$ is $\frac{\pi}{4}$. Since $\tan(\theta) = \frac{\text{Imaginary part}}{\text{Real part}}$, we have: $\tan\left(\frac{\pi}{4}\right) = \frac{\frac{2y}{(x + 1)^2 + y^2}}{\frac{x^2 + y^2 - 1}{(x + 1)^2 + y^2}} = 1$ This simplifies to: $\frac{2y}{(x + 1)^2 + y^2} = \frac{x^2 + y^2 - 1}{(x + 1)^2 + y^2}$ Since $(x + 1)^2 + y^2 \neq 0$ (because $z \neq -1$), we can multiply both sides by it: $2y = x^2 + y^2 - 1$

• Step 4: Convert to Standard Circle Equation Rearrange the equation to the standard form of a circle's equation, $(x - h)^2 + (y - k)^2 = r^2$. $x^2 + y^2 - 2y - 1 = 0$ Complete the square for the $y$ terms: $x^2 + (y^2 - 2y + 1) - 1 - 1 = 0$ $x^2 + (y - 1)^2 = 2$

• Step 5: Identify Center and Radius Comparing $x^2 + (y - 1)^2 = 2$ with $(x - h)^2 + (y - k)^2 = r^2$, we can identify the center and radius:

  • Center: $(h, k) = (0, 1)$
  • Radius: $r = \sqrt{2}$

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs when expanding and simplifying the complex fraction. A small sign error can lead to an incorrect equation.
• Denominator: Remember to exclude $z = -1$ because it makes the denominator zero in the original expression, causing it to be undefined.
• Completing the Square: Practice completing the square to easily convert quadratic equations into the standard circle equation.

Summary

By substituting $z = x + iy$ into the given equation and simplifying, we obtained the Cartesian equation $x^2 + (y - 1)^2 = 2$. This equation represents a circle with center at $(0, 1)$ and radius $\sqrt{2}$.

The final answer is \boxed{(0, 1) and radius \sqrt{2}}, which corresponds to option (A).