Question

If $y_1=max|\mid z-\omega \mid -\mid z-{\omega }^2\mid |$,where $\left|z\right|=2$

and $y_2=max|\mid z-\omega \mid -\mid z-{\omega }^2\mid |$, where $\left|z\right|=\frac{1}{2}$

and $\omega$ and ${\omega }^2$ are complex cube roots of unity, then

• A. $y_1=\sqrt{3},y_2=\sqrt{3}$
• B. $y_1<\sqrt{3},y_2=\sqrt{3}$
• C. $y_1=\sqrt{3},y_2<\sqrt{3}$
• D. none of these

Answer 1

The correct option is C $y_1=\sqrt{3},y_2<\sqrt{3}$

Triangular inequality in $\Delta ZPQ$
$|\mid z-w\mid -\mid z-w^2\mid |\le \left|PQ\right|\le \sqrt{3}$
In case of $\left|z\right|=2$ it is possible to have equality
$max|\mid z-w\mid -\mid z-w^2\mid |=\sqrt{3}$
but in case of $\left|z\right|=\frac{1}{2}$; equality is not possible
$y_2<\sqrt{3}$