QJEEM15Maea39: JEE Main 2015 Mathematics - Class 11 JEEM15Maea39 Solution: A complex number z is said to be unimodular if . Suppose and…

Question

Verified Solution • 2025–26 Session moderate

A complex number z is said to be unimodular if

\,\left| z \right| = 1

. Suppose

{z_1}

and

{z_2}

are complex numbers such that

{{{z_1} - 2{z_2}} \over {2 - {z_1}\overline {{z_2}} }}

is unimodular and

{z_2}

is not unimodular. Then the point

{z_1}

lies on a :

Accepted Answer

\left| {{{{z_1} - 2{z_2}} \over {2 - {z_1}{{\overline z }_2}}}} \right| = 1 \Rightarrow {\left| {{z_1} - 2{z_2}} \right|^2} = {\left| {2 - {z_1}{{\overline z }_2}} \right|^2}

\Rightarrow \left( {{z_1} - 2{z_2}} \right)\left( {\overline {{z_1} - 2{z_2}} } \right)

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,

= \left( {2 - {z_1}{{\overline z }_2}} \right)\left( {\overline {2 - {z_1}{{\overline z }_2}} } \right)

\Rightarrow \left( {{z_1} - 2{z_1}} \right)\left( {{{\overline z }_1} - 2{{\overline z }_2}} \right)

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,

= \left( {2 - {z_1}\overline {{z_2}} } \right)\left( {2 - {{\overline z }_1}{z_2}} \right)

\Rightarrow \left( {{z_1}{{\overline z }_1}} \right) - 2{z_1}{\overline z _2} - 2{\overline z _1}{z_2} + 4{z_2}{\overline z _2}

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,

= 4 - 2{\overline z _1}{z_2} - 2{z_1}{\overline z _2} + {z_1}{\overline z _1}{z_2}{\overline z _2}

\Rightarrow {\left| {{z_1}} \right|^2} + 4{\left| {{z_2}} \right|^2}

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,

= 4 + {\left| {{z_1}} \right|^2}{\left| {{z_2}} \right|^2}

\Rightarrow {\left| {{z_1}} \right|^2} + 4{\left| {{z_2}} \right|^2} - 4 - {\left| {{z_1}} \right|^2}{\left| {{z_2}} \right|^2} = 0

\left( {{{\left| {{z_1}} \right|}^2} - 4} \right)\left( {1 - {{\left| {{z_2}} \right|}^2}} \right) = 0

As

\,\,\,\left| {{z_2}} \right| \ne 1\,\,\,

\therefore

\,\,\,{\left| {{z_1}} \right|^2} = 4 \Rightarrow \left| {{z_1}} \right| = 2

\Rightarrow \,

Point

\,{z_1}\,

lies on circle of radius

2.

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