If a complex number \(z\) statisfies the equation \(x + \sqrt{2}|z + 1| + i = 0\), then \(|z|\) is equal to :
Show Hint
- \(2\)
- \(\sqrt{3}\)
- \(\sqrt{5}\)
- \(1\)
The Correct Option is C
Solution and Explanation
Write \(z = a + ib\). Then \(a + ib = -\sqrt{2}|z+1| - i\).
Comparing imaginary parts: \(b = -1\). Comparing real parts: \(a = -\sqrt{2}|z+1|\), so \(a \le 0\).
\(|z+1|^2 = (a+1)^2 + 1\), so \(a^2 = 2[(a+1)^2 + 1] = 2a^2 + 4a + 4\).
\(a^2 + 4a + 4 = 0\), so \((a+2)^2 = 0\) and \(a = -2\) (which is \(\le 0\)).
\(z = -2 - i\), so \(|z| = \sqrt{4 + 1} = \sqrt{5}\).
Answer: (3) \(\sqrt{5}\)
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