Question

The region represented by {z = x + iy ∈ C : |z| – Re(z) ≤ 1} is also given by the inequality is ______.

Options

• y² ≥ 2(x + 1)

• `y^2 ≤ 2(x + 1/2)`

• `y^2 ≤ x + 1/2`

• y² ≥ x + 1

Answer 1

The region represented by {z = x + iy ∈ C : |z| – Re(z) ≤ 1} is also given by the inequality is `underlinebb(y^2 ≤ 2(x + 1/2))`.

Explanation:

∵ |z| – Re(z) ≤ 1  ...(∵ z = x + iy)

⇒ `sqrt(x^2 + y^2) - x ≤ 1`

⇒ `sqrt(x^2 + y^2) ≤ 1 + x` 

⇒ x² + y² ≤ 1 + x² + 2x

⇒ y² ≤ 1 + 2x

⇒ `y^2 ≤ 2(x + 1/2)`