complex locus(prove collinear) (1 Viewer)

[zcrt]

The points
which represent z1, z2, z3 lie on a circle through the origin. Show that the
points which represent 1/z1, 1/z2, 1/z3 are collinear.


make it detailed thx.

 

[Carrotsticks]

The points
which represent z1, z2, z3 lie on a circle through the origin. Show that the
points which represent 1/z1, 1/z2, 1/z3 are collinear.


make it detailed thx.

There are a couple of available proofs (the most classic of which in textbooks is via circle geometry). But I'll provide an alternative one which demonstrates the usefulness of a substitution.

Let the centre of the circle be C, so |z_k - C| = |C|, where k=1,2,3. Geometrically, this means that the distance of z_k from C is always equal to |C|, which is the radius of the circle.

Make the substitution w=1/z_k, so z_k = 1/w.

|1/w - C| = |C|

|1-Cw| = |C||w|

|1/C - w| = |w|

Geometrically, this is the locus of all points equidistant from the complex number 1/C and the origin, which is a straight line.

So the locus of W is a straight line, but since W=1/z_k, then 1/z_k lies on a straight line, where k=1,2,3.