MathDB

(a) Let

z_1,z_2

be complex numbers such that

z_1z_2=1

and

|z_1-z_2|=2

. Let

A,B,M_1,M_2

denote the points in complex plane corresponding to

-1,1,z_1,z_2

, respectively. Show that

AM_1BM_2

is a trapezoid and compute the lengths of its non-parallel sides. Specify the particular cases. (b) Let

\mathcal C_1

and

\mathcal C_2

be circles in the plane with centers

O_1

and

O_2

, respectively, and with radius

d\sqrt2

, where

2d=O_1O_2

. Let

P

and

Q

be two variable points on

\mathcal C_1

and

\mathcal C_2

respectively, both on

O_1O_2

on on different sides of

O_1O_2

, such that

PQ=2d

. Prove that the locus of midpoints

I

of segments

PQ

is the same as the locus of points

M

with

MO_1\cdot MO_2=m

for some

m

Problem Statement