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  • If a complex number z satisfies the equation <img alt="z+\sqrt{2}\left | z+1 \right |+i=0" src="http://entrancecorner.oncodecogs.com/gif.latex?z&plus;%5Csqrt%7B2%7D%5Cleft%20%7C%20z&pl

If a complex number z satisfies the equation z+\sqrt{2}\left | z+1 \right |+i=0, then \left | z \right | is equal to :

Option: 1

2


Option: 2

\sqrt{3}


Option: 3

\sqrt{5}


Option: 4

1


Answers (1)

best_answer

Given equation is 

z+\sqrt{2}|z+1|+i=0

put z =  x + iy in the above equatiom

\\(x+i y)+\sqrt{2}|x+i y+1|+i=0 \\ \Rightarrow x+i y+\sqrt{2}\left[\sqrt{(x+1)^{2}+y^{2}}\right]+i=0

Now, equating real and imaginary part, we get

\\x+\sqrt{2} \sqrt{(x+1)^{2}+y^{2}}=0 \text { and } \\ y+1=0 \Rightarrow y=-1 \\ \Rightarrow x+\sqrt{2} \sqrt{(x+1)^{2}+(-1)^{2}}=0 \ \quad(\because y=-1) \\ \Rightarrow \sqrt{2} \sqrt{(x+1)^{2}+1}=-x \\ \Rightarrow 2\left[(x+1)^{2}+1\right]=x^{2} \\ \Rightarrow x^{2}+4 x+4=0 \\ \Rightarrow x=-2

 z = -2 - i

|z| = \sqrt{(-2)^2+(-1)^2}=\sqrt5

Posted by

Sanket Gandhi

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