Exams Joint Entrance Examination Mathematics Complex Numbers Question 83 Question 83 of 455 Medium MCQ
Let z 1 , z 2 , z 3 z_1,z_2,z_3 z 1 , z 2 , z 3 be three complex numbers on the circle ∣ z ∣ = 1 |z|=1 ∣ z ∣ = 1 with arg ( z 1 ) = − π 4 \arg(z_1)=-\dfrac{\pi}{4} arg ( z 1 ) = − 4 π , arg ( z 2 ) = 0 \arg(z_2)=0 arg ( z 2 ) = 0 , arg ( z 3 ) = π 4 \arg(z_3)=\dfrac{\pi}{4} arg ( z 3 ) = 4 π . If ∣ z 1 z ˉ 2 + z 2 z ˉ 3 + z 3 z ˉ 1 ∣ 2 = α + β 2 |z_1\bar z_2+z_2\bar z_3+z_3\bar z_1|^2=\alpha+\beta\sqrt{2} ∣ z 1 z ˉ 2 + z 2 z ˉ 3 + z 3 z ˉ 1 ∣ 2 = α + β 2 , α , β ∈ Z \alpha,\beta\in\mathbb Z α , β ∈ Z , then the value of α 2 + β 2 \alpha^2+\beta^2 α 2 + β 2 is:
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\[z_1=e^{-i\pi/4}=\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{2}},\quad z_2=1,\quad z_3=e^{i\pi/4}=\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}.\]
\[\bar z_1=\frac1{\sqrt{2}}+\frac{i}{\sqrt{2}},\quad \bar z_2=1,\quad \bar z_3=\frac1{\sqrt{2}}-\frac{i}{\sqrt{2}}.\]
\[z_1\bar z_2=\frac1{\sqrt{2}}-\frac{i}{\sqrt{2}},\qquad z_2\bar z_3=\frac1{\sqrt{2}}-\frac{i}{\sqrt{2}},\]
\[z_3\bar z_1=\left(\frac1{\sqrt{2}}+\frac i{\sqrt{2}}\right)^2=\frac{(1+i)^2}{2}=\frac{2i}{2}=i.\]
Sum: 1 2 − i 2 + 1 2 − i 2 + i = 2 + i ( 1 − 2 ) \dfrac1{\sqrt{2}}-\dfrac i{\sqrt{2}}+\dfrac1{\sqrt{2}}-\dfrac i{\sqrt{2}}+i = \sqrt{2}+i(1-\sqrt{2}) 2 1 − 2 i + 2 1 − 2 i + i = 2 + i ( 1 − 2 ) .
\[|\sqrt{2}+i(1-\sqrt{2})|^2=2+(1-\sqrt{2})^2=2+1-2\sqrt{2}+2=5-2\sqrt{2}.\]
So α = 5 , β = − 2 \alpha=5,\ \beta=-2 α = 5 , β = − 2 , and α 2 + β 2 = 25 + 4 = 29 \alpha^2+\beta^2=25+4=29 α 2 + β 2 = 25 + 4 = 29 . Answer the question to unlock