The-number-of-solutions-to-the-equation-z-z-2i-2-z-i-is-

Question Number 139116 by EnterUsername last updated on 22/Apr/21

The number of solutions to the equation

z \overset{―}{(z - 2 i)} = 2 (z + i) is ?

Answered by qaz last updated on 22/Apr/21

z \cdot \overset{―}{z - 2 i} = z \bar{z} + z \overset{―}{- 2 i} = z \bar{z} + 2 i z = 2 (z + i)

z = a + b i

a^{2} + b^{2} + 2 i (a + b i) = 2 (a + (b + 1) i)

\Rightarrow a^{2} + b^{2} - 2 b = 2 a

2 a = 2 (b + 1)

\Rightarrow b = \frac{1 + \sqrt{3}}{2}, a = \frac{3 + \sqrt{3}}{2}

o r b = \frac{1 - \sqrt{3}}{2}, a = \frac{3 - \sqrt{3}}{2}

\Rightarrow z = \frac{3 + \sqrt{3}}{2} + \frac{1 + \sqrt{3}}{2} i o r \frac{3 - \sqrt{3}}{2} + \frac{1 - \sqrt{3}}{2} i

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