Let $z_1,z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=-\frac\pi4$, $\arg(z_2)=0$ and $\arg(z_3)=\frac\pi4$. If $|z_1\bar z_2+z_2\bar z_3+z_3\bar z_1|^2=\alpha+\beta\sqrt2$, $\alpha,\beta\in\mathbf Z$, then the value of $\alpha^2+\beta^2$ is :

Question

Dr. Abubacker · Accepted Answer

[Solution by Dr. Abubacker] $z_1\bar z_2=e^{-i\pi/4}$, $z_2\bar z_3=e^{-i\pi/4}$, $z_3\bar z_1=e^{i\pi/2}$. The sum is $\sqrt2+i(1-\sqrt2)$, with modulus squared $2+(1-\sqrt2)^2=5-2\sqrt2$. So $\alpha^2+\beta^2=25+4=29$. Answer (2).

Q. Let $z_1,z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=-\frac\pi4$, $\arg(z_2)=0$ and $\arg(z_3)=\frac\pi4$. If $|z_1\bar z_2+z_2\bar z_3+z_3\bar z_1|^2=\alpha+\beta\sqrt2$, $\alpha,\beta\in\mathbf Z$, then the value of $\alpha^2+\beta^2$ is :

Explanation

Reviewed by Dr. Abubacker

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