Question

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

Options

• A straight line passing through the origin and the first and third quadrants.

• A straight line passing through the origin and the second and fourth quadrants.

• A straight line passing through the point (1, -1) and having a slope of -1.

• A straight line passing through the point (2, 1) and having slope 1/2

Answer 1

A straight line passing through the origin and the first and third quadrants.

Explanation:

Let z = x + iy

|z + 1 − i| = |z − 1 + i|

∴ |x + iy + 1 − i| = |x + iy − 1 + i|

∴ |(x + 1) + i(y − 1)| = |(x − 1) + i(y + 1)|

∴ (x + 1)² + (y − 1)² = (x − 1)² + (y + 1)²

∴ 4x − 4y = 0

∴ x − y = 0

∴ x = y

which is a straight line passing through the origin and lies in the first and third quadrants.