- #1

utkarshakash

Gold Member

- 855

- 13

## Homework Statement

If |z|=1 and |ω-1|=1, where z, ω [itex]\in[/itex] C, then find the range of [itex] |2z-1|^{2}+|2ω-1|^{2}.[/itex]

## Homework Equations

## The Attempt at a Solution

Since |ω-1|=1

Squaring both sides and simplifying

[itex] |ω|^{2}=ω+\overline{ω}[/itex]

Also simplifying the expression given in the question

[itex]6-2(z+\overline{z})-2(ω+\overline{ω})+4|ω|^{2}[/itex]

[itex]6-2(z+\overline{z})+2(ω+\overline{ω})[/itex]

Since [itex](ω+\overline{ω})=-1[/itex]

[itex]4-2(z+\overline{z})[/itex]

Since [itex](z+\overline{z}) = 2Re(z)[/itex]

Now the expression reduces to

[itex] \large 4 \left\{ 1-Re(z) \right\} [/itex]

Since |z|=1

∴Locus of z will be a circle with centre at origin and unit radius. So the max Re(z) can be 1 and min Re(z) can be -1. Substituting these in my expression for max and min I get [0,8] but the answer is [2,18].