Question

If $\left|z\right|=2$ and $\frac{z_1-z_3}{z_2-z_3}=\frac{z-2}{z+2}$.

Then show that $z_1,z_2,z_3$ are vertices of a right-angled triangle.

Answer 1

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