Question

Let $z=x+iy,$ where $x,y \in \mathbb{R}$ and $i^{2}=-1$. If $|z-i|=|z-1|$, then $y=$ ________.

• A. $-x$
• B. $x+1$
• C. $-x-1$
• D. $x+2$
• E. $x$

Hint:

$|z-a|=|z-b|$ represents the perpendicular bisector of the line joining $a$ and $b$.

Answer 1

Answer: (E)

Step 1: Concept
Substitute $z=x+iy$ into the equation and simplify.

Step 2: Meaning
$|x+i(y-1)| = |(x-1)+iy|$.

Step 3: Analysis
$x^2 + (y-1)^2 = (x-1)^2 + y^2 \implies x^2 + y^2 - 2y + 1 = x^2 - 2x + 1 + y^2$.

Step 4: Conclusion
$-2y = -2x \implies y = x$. Final Answer: (E)