Question

State True or False for the following:

The inequality |z – 4| < |z – 2| represents the region given by x > 3.

Options

• True

• False

Answer 1

This statement is True.

Explanation:

Given that: |z – 4| < |z – 2|

Let z = x + yi

⇒ |x + yi – 4| < |x + yi – 2|

⇒ |(x – 4) + yi| < |(x – 2) + yi|

⇒ `sqrt((x - 4)^2 + y^2) < sqrt((x - 2)^2 + y^2)`

⇒ (x – 4)² + y² < (x – 2)² + y²

⇒ (x – 4)² < (x – 2)²

⇒ x² + 16 – 8x < x² + 4 – 4x

⇒ –8x + 4x < –16 + 4

⇒ –4x < –12

⇒ x > 3