If-z-1-z-2-and-z-3-z-4-are-two-pairs-of-conjugate-complex-numbers-then-find-value-of-arg-z-1-z-4-arg-z-2-z-3-

Question Number 49174 by rahul 19 last updated on 04/Dec/18

I f z_{1}, z_{2} a n d z_{3}, z_{4} a r e t w o p a i r s o f

c o n j u g a t e c o m p l e x n u m b e r s, t h e n

f i n d v a l u e o f a r g (\frac{z_{1}}{z_{4}}) + a r g (\frac{z_{2}}{z_{3}}) ?

Commented by rahul 19 last updated on 04/Dec/18

a r g (\frac{z_{1} z_{2}}{z_{4} z_{3}}) = a r g (\frac{∣ z_{1} ∣^{2}}{∣ z_{3} ∣^{2}}) = 0

(∵ z_{2} = {\overline{z}}_{1} \Rightarrow z_{1} z_{2} = ∣ z_{1} ∣^{2} a n d z_{3} z_{4} = ∣ z_{3} ∣^{2})

Answered by MJS last updated on 04/Dec/18

z_{1} = p e^{i α}

z_{2} = p e^{- i α}

z_{3} = q e^{i β}

z_{4} = q e^{- i β}

z_{5} = \frac{z_{1}}{z_{4}} = \frac{p}{q} e^{i (α + β)}

z_{6} = \frac{z_{2}}{z_{3}} = \frac{p}{q} e^{- i (α + β)}

\arg (z_{5}) + \arg (z_{6}) = 0

Commented by rahul 19 last updated on 04/Dec/18

thank you sir!��