Question

The equation $|z + 1 - i| = |z - 1 + i|$ represents a (where z is a complex number)

• A. Straight line passing through the origin and the first and third quadrant.
• B. Straight line passing through the origin and the second and fourth quadrant.
• C. Straight line passing through the point $(1, -1)$ and having slope $-1$ .
• D. Straight line passing through the point $(2, 1)$ and having slope $\frac{1}{2}$.

Hint:

$|z - a| = |z - b|$ is the set of points equidistant from $a$ and $b$. If $a = -b$, the line always passes through the origin.

Answer 1

The Correct Option is A

Step 1: Concept
The equation $|z - z_1| = |z - z_2|$ represents the perpendicular bisector of the segment joining $z_1$ and $z_2$.

Step 2: Meaning
Here, $z_1 = -1 + i$ (Point: $(-1, 1)$) and $z_2 = 1 - i$ (Point: $(1, -1)$).

Step 3: Analysis
The midpoint of $(-1, 1)$ and $(1, -1)$ is $(\frac{-1+1}{2}, \frac{1-1}{2}) = (0, 0)$. The slope of the segment joining them is $m = \frac{-1-1}{1-(-1)} = -1$. The slope of the perpendicular bisector is $m' = -1/m = 1$. The equation is $y - 0 = 1(x - 0) \implies y = x$.

Step 4: Conclusion
$y = x$ is a line passing through the origin and quadrants I and III. Final Answer: (A)