Let-z-be-a-complex-number-If-z-1-z-1-and-arg-z-1-z-1-pi-4-Then-z-is-

Question Number 139118 by EnterUsername last updated on 22/Apr/21

Let z be a complex number . If ∣ z + 1 ∣=∣ z - 1 ∣

and \arg (\frac{z - 1}{z + 1}) = \frac{π}{4} . Then z is ?

Answered by qaz last updated on 22/Apr/21

∣ z + 1 ∣=∣ z - 1 ∣

\Rightarrow z = a + b i = b i

a r g (\frac{b i - 1}{b i + 1}) = a r g (\frac{{(b i - 1)}^{2}}{1 - b^{2}}) = a r g (\frac{1 - b^{2} - 2 i b}{1 - b^{2}})

= \tan^{- 1} \frac{- 2 b}{1 - b^{2}} = \frac{π}{4}

1 - b^{2} = - 2 b

\Rightarrow b = 1 \pm \sqrt{2}

\Rightarrow z = (1 \pm \sqrt{2}) i

Commented by mr W last updated on 22/Apr/21

i t h i n k o n l y z = (1 + \sqrt{2}) i i s s o l u t i o n .

b e c a u s e w i t h z = (1 - \sqrt{2}) i :

a r g (\frac{z - 1}{z + 1}) = \frac{3 π}{4} \neq \frac{π}{4}

Commented by EnterUsername last updated on 22/Apr/21

T h a n k s S i r s