Question

If z , -iz , 1 is collinear show z lies in circle

Answer 1

Solution:

Let, z=x + i y

$|z|=\sqrt{x^2+y^2},\\\\ z^2=x^2+y^2\\\\ |i|=1$, |z|*|z|=|z|²,       |i|²=1, |z*i|=|z||i|

As,it is given that, z, -i z,1 are Collinear.

So, |z+iz|² + |-i z -1|²=|z-1|²

→ |z|²+ |i z|²+ 2 |z| |i z|+|i z|²+1+2 |i z|=|z|²+1+2 |z|

→|z|²+|z|²+2 |i| |z|²+|z|²+1+2 |z| |i|=|z|²+1+2 |z|→→|a b|=|a||b|, and |i|=1

→5|z|²+2 |z|-2 |z|- |z|²-1-2 |z|=0

→4 z²=1

→$Z^2=\frac{1}{4}$

$x^2+y^2=\frac{1}{4}$

Represents a circle having center (0,0) and Radius $\frac{1}{2}$.