If z and w are complex numbers satisfying barz+i barw=0 and amp(z w)=π , then amp(w) is equal to, where amp(w)∈ (- π , π ]

Question

If z and w are complex numbers satisfying barz+i barw=0 and amp(z w)=π , then amp(w) is equal to, where amp(w)∈ (- π , π ] (A) (π /4) (B) (- π /4) (C) (π /2) (D) (3 π /4)

Tardigrade Admin · Accepted Answer

Given overset-z+i overset-w=0⇒ overset-z=-i overset-w or z=iw or (z/w)=i amp(z)-amp(w)=amp i=(π /2)....(i) also amp(z w)=π amp(z)+amp(w)=π ldots ..(i i) Adding (i) &(i i), we get, 2amp(z)=(3 π /2) ⇒ amp(z)=(3 π /4) Also, amp(w)=(π /4)

Q. If $z$ and $w$ are complex numbers satisfying $\overset{z}{ˉ} + i \overset{w}{ˉ} = 0$ and $am p (z w) = π,$ then $am p (w)$ is equal to, where $am p (w) \in (- π, π]$

$\frac{π}{4}$

$\frac{- π}{4}$

$\frac{π}{2}$

$\frac{3 π}{4}$

Q. If z and w are complex numbers satisfying zˉ+iwˉ=0 and amp(zw)=π, then amp(w) is equal to, where amp(w)∈(−π,π]

4π​

4−π​

2π​

43π​

Given z−+iw−=0⇒z−=−iw− or z=iw or wz​=i amp(z)−amp(w)=ampi=2π​....(i) also amp(zw)=π amp(z)+amp(w)=π…..(ii) Adding (i)&(ii), we get, 2amp(z)=23π​ ⇒amp(z)=43π​ Also, amp(w)=4π​

Q. If $z$ and $w$ are complex numbers satisfying $\overset{z}{ˉ} + i \overset{w}{ˉ} = 0$ and $am p (z w) = π,$ then $am p (w)$ is equal to, where $am p (w) \in (- π, π]$

$\frac{π}{4}$

$\frac{- π}{4}$

$\frac{π}{2}$

$\frac{3 π}{4}$