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Question - Let $z_1 , z_2 ,z_3 $ be 3 complex numbers satisfying $|z|=1$ and $$4z_3=3(z_1+z_2)$$ Then find the value of $|z_1-z_2|$.

Looking at the question i could immediately see a geometric solution that we can just rearrange the equation like $$\dfrac{2}{3}z_3=\dfrac{z_1+z_2}{2}$$ Now midpoint of $Z_1Z_2$ divides $OZ_3$ in the ration $2/3$ ($O$ is origin $Z_i$ is the respective vertex).. Using this fact we get answer as $\boxed{\dfrac{2\sqrt{5}}{3}}$

I was wondering if there was any purely algebraic solution to this? I couldnt find any.

Aleph
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$$\left(\frac43\right)^2=|z_1+z_2|^2=2+2\Re(z_1\bar z_2)$$ hence $$|z_1-z_2|^2=2-2\Re(z_1\bar z_2)=4-\left(\frac43\right)^2=\frac{20}9.$$

Anne Bauval
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