If $A\left( \mathcal{z}_{1} \right),\ B\left( \mathcal{z}_{2} \right)$ and $C(\mathcal{z}_{3})$ be the vertices of a triangle $ABC$ in which $\angle ABC = \frac{\pi}{4}$ and $\frac{\ AB}{BC} = \sqrt{2}$, then the value of $\mathcal{z}_{2}$ is equal to

Question

Dr. Abubacker · Accepted Answer

(c)
Given, $\frac{AB}{BC} = \sqrt{2}$
Consider the rotation about $'B^{'}$, we get
$\frac{\mathcal{z}_{1} - \mathcal{z}_{2}}{\mathcal{z}_{3} - \mathcal{z}_{2}} = \frac{|\mathcal{z}_{1} - \mathcal{z}_{2}|}{\mathcal{|z}_{3} - \mathcal{z}_{2}|}\ e^{i\pi/4}$
$= \frac{AB}{BC}\ e^{i\pi/4}$
$= \ \sqrt{2}\left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right) = 1 + i$
$\Rightarrow \ \mathcal{z}_{1} - \mathcal{z}_{2} = (1 + i)\left( \mathcal{z}_{3} - \mathcal{z}_{2} \right)$
$\Rightarrow \ \mathcal{z}_{1} - (1 + i)\mathcal{z}_{3} = \mathcal{z}_{2}(1 - 1 - i)$
$\Rightarrow \ i\mathcal{z}_{2} = - \mathcal{z}_{1} + (1 + i)\mathcal{z}_{3}$
$\Rightarrow \ \mathcal{z}_{2} = i\mathcal{z}_{1} - i(1 + i)\mathcal{z}_{3}$
$= \ \mathcal{z}_{3} + i(\mathcal{z}_{1} - \mathcal{z}_{3})$

Q. If $A\left( \mathcal{z}_{1} \right),\ B\left( \mathcal{z}_{2} \right)$ and $C(\mathcal{z}_{3})$ be the vertices of a triangle $ABC$ in which $\angle ABC = \frac{\pi}{4}$ and $\frac{\ AB}{BC} = \sqrt{2}$, then the value of $\mathcal{z}_{2}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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