Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, the modulus is given by $|z| = \sqrt{x^2 + y^2}$, representing the distance from the origin to the point $(x, y)$ in the complex plane.
• Geometric Interpretation of $|z - z_0| = r$: This equation represents a circle with center $z_0$ and radius $r$ in the complex plane.
• Geometric Interpretation of $|z - z_1| = |z - z_2|$: This equation represents the perpendicular bisector of the line segment joining the points representing the complex numbers $z_1$ and $z_2$ in the complex plane.

Step-by-Step Solution

Step 1: Analyze the first condition, $|z| - 2 = 0$

• Explanation: This condition defines the modulus of the complex number $z$. We will rewrite it to obtain an equation in terms of $x$ and $y$.
• Working: The given condition is: $|z| - 2 = 0$ Rearranging, we get: $|z| = 2$ Substituting $z = x + iy$, we have: $\sqrt{x^2 + y^2} = 2$ Squaring both sides, we obtain: $x^2 + y^2 = 4 \quad \text{(Equation 1)}$
• Geometric Interpretation: This equation represents a circle centered at the origin $(0,0)$ with a radius of $2$.

Step 2: Analyze the second condition, $|z - i| - |z + 5i| = 0$

• Explanation: This condition equates the distances from $z$ to $i$ and from $z$ to $-5i$. We'll rewrite it and interpret it geometrically.
• Working: The given condition is: $|z - i| - |z + 5i| = 0$ Rearranging, we get: $|z - i| = |z + 5i|$ Rewriting $z + 5i$ as $z - (-5i)$, we have: $|z - i| = |z - (-5i)|$ Substituting $z = x + iy$, we get: $|x + iy - i| = |x + iy + 5i|$ $|x + i(y - 1)| = |x + i(y + 5)|$ $\sqrt{x^2 + (y - 1)^2} = \sqrt{x^2 + (y + 5)^2}$ Squaring both sides, we get: $x^2 + (y - 1)^2 = x^2 + (y + 5)^2$ $x^2 + y^2 - 2y + 1 = x^2 + y^2 + 10y + 25$ Subtracting $x^2 + y^2$ from both sides: $-2y + 1 = 10y + 25$ $-12y = 24$ $y = -2 \quad \text{(Equation 2)}$
• Geometric Interpretation: This equation represents a horizontal line at $y = -2$, which is the perpendicular bisector of the line segment joining $(0, 1)$ and $(0, -5)$.

Step 3: Combine both conditions to find $x$ and $y$

• Explanation: We will substitute the value of $y$ from Equation 2 into Equation 1 to find the value of $x$.
• Working: Substitute $y = -2$ into $x^2 + y^2 = 4$: $x^2 + (-2)^2 = 4$ $x^2 + 4 = 4$ $x^2 = 0$ $x = 0$ Thus, the complex number $z$ is $0 - 2i$, so $x = 0$ and $y = -2$.

Step 4: Verify which option is correct

• Explanation: We substitute $x=0$ and $y=-2$ into each of the options to find the correct equation.
• Working:
  • (A) $x + 2y - 4 = 0 \Rightarrow 0 + 2(-2) - 4 = -8 \neq 0$ (False)
  • (B) $x^2 + y - 4 = 0 \Rightarrow 0^2 + (-2) - 4 = -6 \neq 0$ (False)
  • (C) $x + 2y + 4 = 0 \Rightarrow 0 + 2(-2) + 4 = -4 + 4 = 0$ (True)
  • (D) $x^2 - y + 3 = 0 \Rightarrow 0^2 - (-2) + 3 = 5 \neq 0$ (False)

Common Mistakes & Tips

• Sign Errors: Be especially careful with signs when substituting and simplifying equations, particularly when dealing with $z+5i$ which should be treated as $z - (-5i)$.
• Geometric Visualization: Visualizing the complex numbers and their relationships geometrically can greatly simplify the problem and provide valuable intuition.
• Verification: Always verify your final solution by substituting the obtained values back into the original equations and the answer options.

Summary

By interpreting the given conditions geometrically, we found that $|z| = 2$ represents a circle centered at the origin with radius 2, and $|z - i| = |z + 5i|$ represents the perpendicular bisector of the line segment joining $i$ and $-5i$, which simplifies to the line $y = -2$. Solving these equations simultaneously gave us $x = 0$ and $y = -2$. Substituting these values into the options, we found that the correct equation is $x + 2y + 4 = 0$.

The final answer is \boxed{x+2y+4=0}, which corresponds to option (C).