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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.
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)`
⇒ 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.
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