Let z1, z2 and z3 be three complex numbers on the circle |z|=1 with arg⁡z1=−π4,arg⁡z2=0 and arg⁡z3=π4.If z1z¯2+z2z¯3+z3z¯12=α+β2,α,β∈Z, then the value of α2+β2 is :

Z1=e−iπ4,Z2=1,Z3=eiπ4z1z¯2+z2z¯3+z3z¯12=e−iπ4×1+1×e−iπ4+eiπ4×eiπ42=e−iπ4+e−iπ4+eiπ42=2e−iπ4+i2==|2−2i+i|2=(2)2+(1−2)2=2+1+2−22=5−22α=5,β=−2⇒α2+β2=29

Let z1, z2 and z3 be three complex numbers on the circle |z|=1 with arg⁡z1=−π4,arg⁡z2=0 and arg⁡z3=π4.If z1z¯2+z2z¯3+z3z¯12=α+β2,α,β∈Z, then the value of α2+β2 is :

Z1=e−iπ4,Z2=1,Z3=eiπ4z1z¯2+z2z¯3+z3z¯12=e−iπ4×1+1×e−iπ4+eiπ4×eiπ42=e−iπ4+e−iπ4+eiπ42=2e−iπ4+i2==|2−2i+i|2=(2)2+(1−2)2=2+1+2−22=5−22α=5,β=−2⇒α2+β2=29

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Let z₁, z₂ and z₃ be three complex numbers on the circle $| z | = 1 with \arg (z_{1}) = \frac{- π}{4}, \arg (z_{2}) = 0 and \arg (z_{3}) = \frac{π}{4} .$ If ${|z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}|}^{2} = α + β \sqrt{2}, α, β \in Z,$ then the value of $α^{2} + β^{2}$ is :

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$\begin{aligned} Z_{1} = e^{- i π 4}, Z_{2} = 1, Z_{3} = e^{i π 4} \end{aligned}$

$\begin{array}{r} {|z_{1} {\bar{z}}_{2} + z_{2} {\bar{z}}_{3} + z_{3} {\bar{z}}_{1}|}^{2} = {|e^{- i \frac{π}{4}} \times 1 + 1 \times e^{- i \frac{π}{4}} + e^{i \frac{π}{4}} \times e^{i \frac{π}{4}}|}^{2} \end{array}$

$= \begin{array}{r} {|e^{- i \frac{π}{4}} + e^{- i \frac{π}{4}} + e^{i \frac{π}{4}}|}^{2} \end{array}$

$\begin{array}{r} = {|2 e^{- i \frac{π}{4}} + i|}^{2} == | \sqrt{2} - \sqrt{2} i + i |^{2} \end{array}$

$\begin{array}{r} = (\sqrt{2})^{2} + (1 - \sqrt{2})^{2} = 2 + 1 + 2 - 2 \sqrt{2} = 5 - 2 \sqrt{2} \end{array}$

$\begin{array}{r} α = 5, β = - 2 \end{array}$

$\Rightarrow α^{2} + β^{2} = 29$

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Let z1, z2 and z3 be three complex numbers on the circle |z|=1 with arg⁡z1=−π4,arg⁡z2=0 and arg⁡z3=π4.If z1z¯2+z2z¯3+z3z¯12=α+β2,α,β∈Z, then the value of α2+β2 is :

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