Two conjugate complex numbers z1 and z1¯ represent two adjacent vertices of a polygon of n sides, whose centre is origin. If Im(z1)Re(z¯1)=2−1 then ratio of the areas of the circles inscribed and circumscribed the polygon is _______

r=|z1|cosπ8R=|z1|ratio=cos2π8

r=|z1|cosπ8R=|z1|ratio=cos2π8

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Two conjugate complex numbers $z_{1}$ and $\bar{z_{1}}$ represent two adjacent vertices of a polygon of n sides, whose centre is origin. If $\frac{Im (z_{1})}{Re ({\bar{z}}_{1})} = \sqrt{2} - 1$ then ratio of the areas of the circles inscribed and circumscribed the polygon is _______

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$\frac{\sqrt{2} - 1}{2}$

$\frac{\sqrt{2} - 1}{\sqrt{2}}$

$\frac{\sqrt{2} + 1}{2 \sqrt{2}}$

$\frac{\sqrt{2} - 1}{2 \sqrt{2}}$

answer is C.

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$\begin{array}{l} r = | z_{1} | \cos \frac{π}{8} \\ R = | z_{1} | \\ r a t i o = \cos^{2} \frac{π}{8} \end{array}$

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