Q. 74.8( 9 Votes )

If |z + i| = |z – i|, prove that z is real.

Answer :

Let z = x + iy

Consider, |z + i| = |z – i|


|x + iy + i| = |x + iy – i|


|x + i(y +1)| = |x + i(y – 1)|





Squaring both the sides, we get


x2 + y2 + 1 + 2y = x2 + y2 + 1 – 2y


x2 + y2 + 1 + 2y – x2 – y2 – 1 + 2y = 0


2y + 2y = 0


4y = 0


y = 0


Putting the value of y in eq. (i), we get


z = x + i(0)


z = x


Hence, z is purely real.


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