Question

The set \( S = \{ z \in \mathbb{C} : |z + 1 - i| = 1 \} \) represents:

• A. the circle with centre at \( (-1, 1) \) and radius 1 unit
• B. the circle with centre at \( (1, -1) \) and radius 1 unit
• C. the closed circular disc with centre at \( (-1, -1) \) and radius 1 unit
• D. the closed circular disc with centre at \( (1, -1) \) and radius 1 unit

Hint:

For equations of the form \( |z - z_0| = r \), the locus of points represents a circle with centre \( z_0 \) and radius \( r \) in the complex plane.

Answer 1

The Correct Option is A

We are given the set: \[ S = \{ z \in \mathbb{C} : |z + 1 - i| = 1 \}. \] This represents the set of complex numbers \( z \) such that the distance from \( z \) to the point \( (-1, 1) \) in the complex plane is 1 unit. This is the equation of a circle in the complex plane. To clarify, let's rewrite the equation: \[ |z + 1 - i| = 1. \] Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, the equation becomes: \[ | (x + 1) + i(y - 1) | = 1. \] The magnitude of a complex number \( a + ib \) is given by \( \sqrt{a^2 + b^2} \). Therefore: \[ \sqrt{(x + 1)^2 + (y - 1)^2} = 1. \] Squaring both sides: \[ (x + 1)^2 + (y - 1)^2 = 1. \] This is the equation of a circle with centre at \( (-1, 1) \) and radius 1 unit. Thus, the correct answer is: \[ \boxed{\text{the circle with centre at } (-1, 1) \text{ and radius 1 unit}}. \]

Answer 2

The set \( S = \{ z \in \mathbb{C} : |z + 1 - i| = 1 \} \) can be interpreted as:

Step 1: Let \( z = x + iy \), where \( x, y \in \mathbb{R} \).
Step 2: Rewrite the expression inside the modulus:
\[ z + 1 - i = (x + iy) + 1 - i = (x + 1) + i(y - 1) \]

Step 3: The condition \( |z + 1 - i| = 1 \) means:
\[ \sqrt{(x + 1)^2 + (y - 1)^2} = 1 \]

Step 4: This is the equation of a circle with center at \( (-1, 1) \) and radius 1.

Therefore, the set \( S \) represents:
\[ \boxed{\text{the circle with centre at } (-1, 1) \text{ and radius } 1 \text{ unit}} \]