Question

If z, -iz, 1 are collinear then prove that z lies on a circle

Answer 1

Answer:

It is given that , z, -i z, 1 are collinear.

Let, z= x+ i y

-i z=-i(x+i y)=- i x-i²y=-i x +y, as i²= -1

1=1 + 0 i

As, points are collinear, so area of triangle will be equal to 0.

Or, Distance between z and 1 is equal to sum of distances between z and -i z and -i z and 1.

So, we adopt the first procedure to prove that the points , z , -i z, 1 are collinear.

Proved below.

The equation of the circle passing through, z ,-i z, 1 is

x²+y²-2 y=0

$|z|^2+(i z+\bar{i z})=0, |z|=\sqrt{x^2+y^2}, i z=i x -y, \bar(iz)=-y- ix$

Hence Proved.