Question

The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.

Options

• Circle x² + y² = 1

• The x-axis

• The y-axis

• The line x + y = 1.

Answer 1

The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on the x-axis.

Explanation:

Given that: `|(i + z)/(i - z)|` = 1

Let z = x + yi

∴ `|(i + x + yi)/(i - x - yi)|` = 1

⇒ `|(x + (y + 1)i)/(-x - (y - 1)i)|` = 1

⇒ `|x + (y + 1)i| = |-x - (y - 1)i|`

⇒ `sqrt(x^2 + (y + 1)^2) = sqrt(x^2 + (y - 1)^2)`

⇒ x² + (y + 1)² = x² + (y – 1)²

⇒ (y + 1)² = (y – 1)²

⇒ y² + 2y + 1 = y² – 2y + 1

⇒ 2y = –2y

⇒ 4y = 0

⇒ x-axis.