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Geometrical Representation of a Complex Number

If z1, z2, z3...

Question

If $z_{1}, z_{2}, z_{3}$ be the vertices of an rightangled isosceles triangle and which is right angled at $z_{2}$ then $z_{1}^{2} + z_{3}^{2} + 2 z_{2}^{2} = k z_{2} (z_{1} + z_{3})$ where $k =$

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Solution

From above diagram $| z_{1} - z_{2} | = | z_{3} - z_{2} | = r$
Now using the rotation of point $C (z_{3})$ about point $B (z_{2})$ will be come $A (z_{1})$
Hence $\frac{z_{1} - z_{2}}{z_{3} - z_{2}} = e^{i π / 2}$
$\Rightarrow z_{1} - z_{2} = i (z_{3} - z_{2})$
$\Rightarrow z_{1}^{2} + z_{2}^{2} - 2 z_{1} z_{2} = - z_{3}^{2} - z_{2}^{2} + 2 z_{3} z_{2} \Rightarrow z_{1}^{2} + z_{3}^{2} + 2 z_{2}^{2} = 2 z_{2} (z_{1} + z_{3})$
$∴ k = 2$

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