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 (B)  (π /2) (C)  (3π /2) (D) π

Tardigrade Admin · Accepted Answer

We have z2= overlinez1 and z4= overlinez3 ∴ z1z2=z1z1=|z1|2 z3z4=z3 overlinez3=|z3|2 Now arg. ((z1/z4))+arg((z2/z3))=arg.((z1z2/z4z3)) =arg.((|z1|2/|z3|2))=arg. ((|z1|2/|z3|))=0 [ ∵ argument of a +ve real number is zero]

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

$0$

$\frac{π}{2}$

$\frac{3 π}{2}$

$π$

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

0

2π​

23π​

π

We have z2​=z1​ and z4​=z3​ ∴z1​z2​=z1​z1​=∣z1​∣2 z3​z4​=z3​z3​=∣z3​∣2 Now arg.(z4​z1​​)+arg(z3​z2​​)=arg.(z4​z3​z1​z2​​) =arg.(∣z3​∣2∣z1​∣2​)=arg.(∣z3​∣∣z1​∣2​)=0 [ ∵ argument of a+ve real number is zero]

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

$0$

$\frac{π}{2}$

$\frac{3 π}{2}$

$π$