If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then arg ((z1/z4))+ arg ((z2/z3)) equals

Question

If z1, z2 and z3, z4 are two pairs of conjugate complex numbers, then arg ((z1/z4))+ arg ((z2/z3)) equals (A) 0, π (B) π,-π (C) (π/2), (3 π/2) (D) 0,2 π

Tardigrade Admin · Accepted Answer

arg ((z1/z4))+ arg ((z2/z3)) = arg ((z1 z2/z4 z3))+2 k π= arg ((|z1|2/|z3|2))+2 k π=2 k π where k=0 or 1 [∵ z1=z3=i., gives answer 2 π and z1=z3=1, gives answer 0]

Q. If $z_{1}, z_{2}$ and $z_{3}, z_{4}$ are two pairs of conjugate complex numbers, then $ar g (\frac{z _{1}}{z _{4}}) + ar g (\frac{z _{2}}{z _{3}})$ equals

$0, π$

$π, - π$

$\frac{π}{2}, \frac{3 π}{2}$

$0, 2 π$

$ar g (\frac{z _{1}}{z _{4}}) + ar g (\frac{z _{2}}{z _{3}})$
$= ar g (\frac{z _{1} z _{2}}{z _{4} z _{3}}) + 2 kπ = ar g (\frac{∣ z _{1} ∣ ^{2}}{∣ z _{3} ∣ ^{2}}) + 2 kπ = 2 kπ$
where $k = 0$ or 1
$[∵ z_{1} = z_{3} = i$ , gives answer $2 π$ and $z_{1} = z_{3} = 1$ , gives answer 0]

Q. If z1​,z2​ and z3​,z4​ are two pairs of conjugate complex numbers, then arg(z4​z1​​)+arg(z3​z2​​) equals

0,π

π,−π

2π​,23π​

0,2π