Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, the modulus is given by $|z| = \sqrt{x^2 + y^2}$, which represents the distance of the point $(x, y)$ from the origin in the complex plane.
• Geometric Interpretation of Modulus: $|z_1 - z_2|$ represents the distance between the points corresponding to the complex numbers $z_1$ and $z_2$ in the complex plane.
• Perpendicular Bisector: The locus of a point equidistant from two fixed points is the perpendicular bisector of the line segment joining the two fixed points.

Step-by-Step Solution

Step 1: Represent the complex number z in Cartesian form Let $z = x + iy$, where $x$ and $y$ are real numbers. This allows us to work with real coordinates in the complex plane. $z = x + iy$

Step 2: Substitute z into the given equation Substitute $z = x + iy$ into the equation $|z - i| = |z - 1|$: $|x + iy - i| = |x + iy - 1|$ This replaces the complex variable $z$ with its real and imaginary components.

Step 3: Group the real and imaginary parts Group the real and imaginary terms within the modulus: $|x + i(y - 1)| = |(x - 1) + iy|$ This step prepares the expressions for applying the modulus formula.

Step 4: Apply the modulus formula Apply the modulus formula $|a + ib| = \sqrt{a^2 + b^2}$ to both sides: $\sqrt{x^2 + (y - 1)^2} = \sqrt{(x - 1)^2 + y^2}$ This converts the complex equation into a real equation involving square roots.

Step 5: Square both sides Square both sides of the equation to eliminate the square roots: $x^2 + (y - 1)^2 = (x - 1)^2 + y^2$ This simplifies the equation and removes the square roots.

Step 6: Expand and simplify Expand the squared terms: $x^2 + y^2 - 2y + 1 = x^2 - 2x + 1 + y^2$ Now, simplify by cancelling out common terms on both sides ($x^2$, $y^2$, and 1): $-2y = -2x$

Step 7: Solve for y Divide both sides by -2 to find the relationship between $x$ and $y$: $y = x$ This is the equation of the locus.

Step 8: Interpret the equation The equation $y = x$ represents a straight line passing through the origin (0, 0) with a slope of 1.

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs when grouping real and imaginary terms, and when expanding squared terms.
• Geometric Intuition: Always try to visualize the problem geometrically. The equation $|z - a| = |z - b|$ always represents the perpendicular bisector of the line segment joining the points representing $a$ and $b$.
• Modulus Formula: Ensure you apply the modulus formula correctly.

Summary

The equation $|z - i| = |z - 1|$ represents the locus of points equidistant from the complex numbers $i$ and $1$ in the complex plane. By substituting $z = x + iy$ and simplifying, we find the equation of the locus to be $y = x$, which is a straight line passing through the origin with a slope of 1. This corresponds to option (D).

Final Answer The final answer is \boxed{D}, which corresponds to option (D).