If z1 and z2 both satisfy z+ barz=2|z-1| , arg (z1-z2)=(π/4) , then find Im (z1+z2) .

Question

If z1 and z2 both satisfy z+ barz=2|z-1| , arg (z1-z2)=(π/4) , then find Im (z1+z2) . (A) 4 (B) 2 (C) 1 (D) 3

Tardigrade Admin · Accepted Answer

Let z = x + iy , z1 =x1 +iy1 and z2=x2+iy2, z+ barz=2|z-1| ⇒ (x+iy)+(x-iy) =2|x-1+iy| ⇒ 2x=1+y2 ...(i) Since z1 and z2 both satisfy (i), we have 2x1=1+y12 and 2x2=1+y22 ⇒ 2(x1-x2)=(y1+y2)(y1-y2) ⇒ 2=(y1+y2)((y1-y2/x1-x2)) ldots(ii) Again z1-z2=(x1-x2)+i(y1-y2) Therefore, tan θ=(y 1-y2/x1-x2) , where θ=arg(z1-z2) ⇒ tan (π/4)=(y1-y2/x1-x2) (since θ(π/4)) i.e., 1=(y1-y2/x1-x2) From (ii) , we get 2=y1+y2 , i.e., Im (z1+z2)=2

Q. If $z_{1}$ and $z_{2}$ both satisfy $z + \overset{z}{ˉ} = 2 ∣ z - 1 ∣$ , arg $(z_{1} - z_{2}) = \frac{π}{4}$ , then find Im $(z_{1} + z_{2})$ .

$4$

$2$

$1$

$3$

Q. If z1​ and z2​ both satisfy z+zˉ=2∣z−1∣ , arg (z1​−z2​)=4π​ , then find Im (z1​+z2​) .

4

2

1

3

Q. If $z_{1}$ and $z_{2}$ both satisfy $z + \overset{z}{ˉ} = 2 ∣ z - 1 ∣$ , arg $(z_{1} - z_{2}) = \frac{π}{4}$ , then find Im $(z_{1} + z_{2})$ .

$4$

$2$

$1$

$3$