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Z1 &Z2 belongs to complex numbers,(Z1*Z1)+(Z2*Z2) belongs to real number,Z1(Z1*Z1-3Z2)=2 and Z2(3Z1*Z1-Z2*Z2)=1 then value of Z1*Z1+Z2*Z2?

Z1 &Z2 belongs to complex numbers,(Z1*Z1)+(Z2*Z2) belongs to real number,Z1(Z1*Z1-3Z2)=2 and Z2(3Z1*Z1-Z2*Z2)=1 then value of Z1*Z1+Z2*Z2?

Grade:11

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
8 years ago
Ans:

Hello Student,
Please find answer to your question below

z_{1} = x_{1} + iy_{1}
z_{2} = x_{2} + iy_{2}
z_{1}^{2}+z_{2}^{2} = (x_{1}+iy_{2})^{2}+(x_{2} + iy_{2})^{2}
z_{1}^{2}+z_{2}^{2} = (x_{1}^{2}+x_{2}^{2}-y_{1}^{2}-y_{2}^{2})+2i(x_{1}y_{1}+x_{2}y_{2})
Since this is a real number
x_{1}y_{1}+x_{2}y_{2} = 0
(x_{1}+iy_{1})(x_{1}^{2}-y_{1}^{2}+2ix_{1}y_{1}-3(x_{2}+iy_{2})) = 2
x_{1}(x_{1}^{2}-y_{1}^{2}-3x_{2})-y_{1}(2x_{1}y_{1}-3y_{2}) = 2
Similarly you can solve the 2ndequation.
From here you will have the value of z12+ z22.

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