Complex problem solving q (1 Viewer)

gamja · Jan 5, 2023

I swear ive done something with mod=1 before, but cant remember... Any clues on how to solve? thx!

nathanzhou1234 · Jan 5, 2023

gamja said:
View attachment 37345

I swear ive done something with mod=1 before, but cant remember... Any clues on how to solve? thx!

Try geometrically cos z1-z2 represents a the vector between the two. I think its gonna be a parallelogram or smth and thus (z1-z2) and (z1+z2) will be diagonals. Cos they’re equal in magnitude it has to be a rectangle i think.
go shore

hogzillaAnson · Jan 6, 2023

A more algebraic approach in case the visualisations don't kick in during the exam.
$\begin{align*} w &= \frac{z_1}{z_2} \\ z_1 &= wz_2 \\ 1 &= \left|\frac{z_1+z_2}{z_1-z_2}\right|\ [\text{Given}]\\ &= \left|\frac{wz_2+z_2}{wz_2-z_2}\right| \\ &= \left|\frac{\cancel{z_2}(w+1)}{\cancel{z_2}(w-1)}\right| \\ |w+1|&=|w-1|\ [\text{Rearranging}]\\ \text{Let}\ w &= x+yi\ [x,y\in \mathbb{R}]\\|(x+1) + yi|^2 &= |(x-1)+yi|^2 \\ (x+1)^2 + \cancel{y^2} &= (x-1)^2 + \cancel{y^2} \\ \cancel{x^2} + 2x + \cancel{1} &= \cancel{x^2} - 2x + \cancel{1} \\ \therefore x &= 0\ \text{i.e}\ \operatorname{Re} w = 0\end{align*}$

Complex problem solving q (1 Viewer)

gamja

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nathanzhou1234

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