If `omega` is a complex cube root of unity, then the equation `|z- omega|^2+|z-omega^2|^2=lambda` will represent a circle, if

Explanation

`|z-w|^2+|z-w^2|^2=lambda`......(1)

`barz xx z=|z|^2`

so (1) turns into =>`bar|z-w|xx |z-w| +bar|z-w^2| xx |z-w^2|=lamda`

`|barz-barw|xx |z-w| +|barz-bar(w^2)| xx |z-w^2|=lamda`

opening and simplifying we get=>`2zbarz-(w+w^2)barz-(barw+bar(w^2))z+w^2+(w^2)^2=lamda`

`2|z|^2-(w+w^2)barz-(barw+bar(w^2))z+|w^2|+(|w^2|)^2=lamda`

`|w|=1 and 1+w+w^2=0`

substituting these we get =>`2|z|^2+barz+z+2=lamda`

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