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प्रश्न
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
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उत्तर
Let \[z = x + iy\]
Then ,
\[z^2 = \left( x + iy \right)^2 \]
\[ = x^2 + i^2 y^2 + 2ixy\]
\[ = x^2 - y^2 + 2ixy [ \because i^2 = - 1]\]
and
\[\left| z \right| = \sqrt{x^2 + y^2}\]
According to the question,
\[Re\left( z^2 \right) = 0 \text { and } \left| z \right| = 2\]
\[ \Rightarrow x^2 - y^2 = 0 \text { and } \sqrt{x^2 + y^2} = 2\]
\[ \Rightarrow x^2 - y^2 = 0 \text { and } x^2 + y^2 = 4\]
\[\text { On Adding both the equations, we get }\]
\[2 x^2 = 4\]
\[ \Rightarrow x^2 = 2\]
\[ \Rightarrow x = \pm \sqrt{2}\]
\[ \Rightarrow y^2 = 2\]
\[ \Rightarrow y = \pm \sqrt{2}\]
Thus,
\[x = \pm \sqrt{2} \text { and } y = \pm \sqrt{2}\]
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