Determinate the geometric aspect described by M(z) in complex plane such that: arg(z^− −3+i)≡(π/4)[2π]


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Question Number 122659 by mathocean1 last updated on 18/Nov/20

$D e t e r m i n a t e t h e g e o m e t r i c$ $a s p e c t d e s c r i b e d b y M (z) i n$ $c o m p l e x p l a n e s u c h t h a t :$ $a r g (\bar{z} - 3 + i) \equiv \frac{π}{4} [2 π]$
	Answered by MJS_new last updated on 18/Nov/20
	$arg (x) ≡(π/4)[2π] ⇔ x=c+ci let z=a+bi ⇒ z^− =a−bi a−bi−3+i=c+ci ⇒ { ((a−c−3=0 ⇒ c=a−3)),((−b−c+1=0 ⇒ c=1−b)) :} a−3=1−b ⇒ b=4−a z=a+(4−a)i this is a straight line.$
	$\arg (x) \equiv \frac{π}{4} [2 π] \Leftrightarrow x = c + c i$ $let z = a + b i \Rightarrow \bar{z} = a - b i$ $a - b i - 3 + i = c + c i$ $\Rightarrow {\begin{cases} a - c - 3 = 0 \Rightarrow c = a - 3 \\ - b - c + 1 = 0 \Rightarrow c = 1 - b \end{cases}$ $a - 3 = 1 - b \Rightarrow b = 4 - a$ $z = a + (4 - a) i$ $this is a straight line .$
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