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Question

If $$\left|z\right| = 2$$ and $$\displaystyle \frac{z_1 - z_3}{z_2 -z_3} = \frac{z-2}{z+2}$$.
Then show that $$z_1, z_2, z_3$$ are vertices of a right-angled triangle. 


Solution

For circle $$\left|z\right| =2$$, points $$z= 2$$ and $$z= -2$$ are end points of diameter. Therefore,
$$arg\left(\dfrac{z-2}{z+2}\right) = \pm \dfrac{\pi}{2}$$
$$\Longrightarrow arg\left(\dfrac{z_1-z_3}{z_2-z_3}\right) = \pm \dfrac{\pi}{2}$$
Hence, $$z_1, z_2, z_3$$ are vertices of a right-angled triangle.
Ans: 1

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