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4-1.Complex numbers
easy
The values of $z$for which $|z + i|\, = \,|z - i|$ are
A
Any real number
B
Any complex number
C
Any natural number
D
None of these
Solution
(a) Let $z = x + iy$……$(i)$
Given $|z + i|\, = \,|z – i|$
or $|x + iy + i|\, = \,|x + iy – i|$
or $|x + i(y + 1)|\, = \,|x + i(y – 1)|$
or $\sqrt {{x^2} + \,{{(y + 1)}^2}} = \sqrt {{x^2} + {{(y – 1)}^2}} $
or ${x^2} + {(y + 1)^2} = {x^2} + {(y – 1)^2}$
or ${y^2} + 2y + 1 = {y^2} – 2y + 1$or $4y = 0$or $y = 0$
Hence from $(i)$, we get $z = x$, where $x$ is any real number.
Standard 11
Mathematics
