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MCQ

Fill in the Blanks

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

#### Options

Circle x

^{2}+ y^{2}= 1The x-axis

The y-axis

The line x + y = 1.

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#### Solution

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)`

⇒ x^{2} + (y + 1)^{2} = x^{2} + (y – 1)^{2}

⇒ (y + 1)^{2} = (y – 1)^{2}

⇒ y^{2} + 2y + 1 = y^{2} – 2y + 1

⇒ 2y = –2y

⇒ 4y = 0

⇒ x-axis.

Concept: Algebraic Operations of Complex Numbers

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