Let α=8-14 i , A = z ∈ C: (α z - barα overline z / z 2-( overline z )2-112 i )=1 and B= z ∈ C :|z+3 i|=4 Then displaystyle∑z ∈ A ∩ B( operatornameRe z- operatornameIm z) is equal to

Question

Let α=8-14 i , A = z ∈ C: (α z - barα overline z / z 2-( overline z )2-112 i )=1 and B= z ∈ C :|z+3 i|=4 Then displaystyle∑z ∈ A ∩ B( operatornameRe z- operatornameIm z) is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

α=8-14 i z=x+i y a z=(8 x+14 y)+i(-14 x+8 y) z + overline z =2 x z - overline z =2 iy Set A: (2 i(-14 x+8 y)/i(4 x y-112))=1 (x-4)(y+7)=0 x=4 text or y=-7 Set B: x2+(y+3)2=16 when x=4 y=-3 when y=-7 x=0 ∴ A ∩ B = 4-3 i , 0-7 i So, displaystyle∑z ∈ A ∩ B( textRez- textIm z )=4-(-3)+(0-(-7))=14

Q. Let α=8−14i,A={z∈C:z2−(z)2−112iαz−αˉz​=1} and B={z∈C:∣z+3i∣=4} Then z∈A∩B∑​(Rez−Imz) is equal to

α=8−14i z=x+iy az=(8x+14y)+i(−14x+8y) z+z=2xz−z=2iy Set A: i(4xy−112)2i(−14x+8y)​=1 (x−4)(y+7)=0 x=4 or y=−7 Set B: x2+(y+3)2=16 when x=4y=−3 when y=−7x=0 ∴A∩B={4−3i,0−7i} So, z∈A∩B∑​(Rez−Imz)=4−(−3)+(0−(−7))=14

Q. Let $α = 8 - 14 i, A = {z \in C : \frac{α z - α ˉ z}{z ^{2} - ( z ) ^{2} - 112 i} = 1}$ and $B = {z \in C : ∣ z + 3 i ∣ = 4}$ Then $z \in A \cap B \sum (Re z - Im z)$ is equal to