Let's solve the equation \(|z + 1 - i| = |z - 1 + i|\) to understand what geometric shape it represents. Here, \(z\) is a complex number expressed as \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit.
The expression \( |z + 1 - i| \) is the distance from the point \(z\) to the point \((-1, 1)\) on the complex plane. Similarly, \( |z - 1 + i| \) is the distance from \(z\) to the point \( (1, -1) \).
The given equation states that the distance from \(z\) to \((-1, 1)\) is equal to the distance from \(z\) to \( (1, -1) \). Geometrically, this is the definition of the perpendicular bisector of the line segment joining the points \((-1, 1)\) and \( (1, -1) \).
The midpoint of these points is:
\(\left( \frac{-1 + 1}{2}, \frac{1 - 1}{2} \right) = (0, 0)\)
The slope of the line segment joining \((-1, 1)\) and \( (1, -1) \) is:
\(\frac{-1 - 1}{1 + 1} = -1\)
The perpendicular bisector of a line segment will have a slope that is the negative reciprocal of \(-1\), which is \(1\). Hence, the perpendicular bisector is a line passing through the origin \((0, 0)\) with a slope of \(1\), represented by the equation:
\(y = x\)
This line passes through the origin and lies in the first and third quadrants. Therefore, the correct answer is:
Straight line passing through the origin and the first and third quadrant.