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Question
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
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Solution
Given that: z + `sqrt(2) |(z + 1)| + i` = 0
Let z = x + yi
∴ `(x + yi) + sqrt(2)|(x + yi + 1)| + i` = 0
⇒ `x + (y + 1)i + sqrt(2)|(x + 1) + yi|` = 0
⇒ `x + (y + 1)i + sqrt(2) sqrt((x + 1)^2 + y^2)` = 0
⇒ `x + (y + 1)i + sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` = 0 + 0i
⇒ `x + sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` = 0, y + 1 = 0
⇒ x = `- sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` and y = –1
⇒ x2 = 2(x2 + 2x + 1 + y2)
⇒ x2 = 2x2 + 4x + 2 + 2y2
⇒ x2 + 4x + 2 + 2y2 = 0
⇒ x2 + 4x + 2 + 2(–1)2 = 0 .....[∵y = –1]
⇒ x2 + 4x + 4 = 0
⇒ (x + 2)2 = 0
⇒ x + 2 = 0
⇒ x = –2
Hence, z = x + yi = –2 – i.
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