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Question
If \[\left| z - 5i \right| = \left| z + 5i \right|\] , then find the locus of z.
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Solution
\[\left| z - 5i \right| = \left| z + 5i \right|\]
\[ \Rightarrow \left| z - 5i \right|^2 = \left| z + 5i \right|^2 \]
\[ \Rightarrow \left( z - 5i \right)\left( \bar{{z - 5i}} \right) = \left( z + 5i \right)\left( \bar{{z + 5i}} \right) \left[ \because z \bar{z} = \left| z \right|^2 \right]\]
\[ \Rightarrow \left( z - 5i \right)\left( \bar{z} + 5i \right) = \left( z + 5i \right)\left( \bar{z} - 5i \right)\]
\[ \Rightarrow z \bar{z} + 5zi - 5 \bar{z}i - 25 i^2 = z \bar{z} - 5zi + 5 \bar{z}i - 25 i^2 \]
\[ \Rightarrow 5zi + 5zi = 5 \bar{z}i + 5 \bar{z}i\]
\[ \Rightarrow 10zi = 10 \bar{z}i\]
\[ \Rightarrow z = \bar{z}\]
\[ \Rightarrow \text{z is purely real }\]
Hence, the locus of z is real axis.
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