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Question
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
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Solution
Given that: `|(z - 5i)/(z + 5i)|` = 1
Let z = x + yi
∴ `|(x + yi - 5i)/(x + yi + 5i)|` = 1
⇒ `|(x + (y - 5)i)/(x + (y + 5)i)|` = 1
⇒ `|x + (y - 5)i| = |x + (y + 5)i|`
⇒ `x^2 + (y - 5)^2 = x^2 + (y + 5)^2`
⇒ `(y - 5)^2 = (y + 5)^2`
⇒ y2 + 25 – 10y = y2 + 25 + 10y
⇒ 20y = 0
⇒ y = 0
Hence, z lies on x-axis i.e., real axis.
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