CircleHard
Question
A complex number z is said to be unimodular if |z| = 1. Suppose z1 and z2 are complex numbers such that 
is unimodular and z2 is not unimodular. Then the point z1 lies on a :
is unimodular and z2 is not unimodular. Then the point z1 lies on a :Options
A.circle of radius 2
B.circle of radius √2
C.straight line parallel to x-axis
D.straight line parallel to y-axis
Solution
= 1 ⇒ |z1 − 2z2|2 = |2 − z1
|2⇒ (z1 − 2z2)(
− 
) = (2 − z1)(2 − z2)⇒ |z1|2 + 4|z2|2 − 4 − |z1|2 |z2|2 = 0
⇒ 4(|z2|2 − 1) − |z1|2 (|z2|2 − 1) = 0
⇒ |z1|2 − 4 = 0⇒|z1| = 2 is a circle of radius 2 and centre at origin.
Create a free account to view solution
View Solution FreeMore Circle Questions
The locus of centre of the circle which cuts the circle x2 + y2 = k2 orthogonally and passes through the point (p,q) is ...A variable circle passes through the fixed point A (p, q) and touches x-axis. The locus of the other end of the diameter...The locus of a point P(α, β) moving under the condition that the line y = αx + β a tangent to the hy...The angle between the two tangents from the origin to the circle (x -7)2 + (y + 1)2 = 25 equals -...The straight line (x − 2) + (y + 3) = 0 cuts the circle (x − 2)2 + (y − 3)2 = 11 at-...