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If $z = \sqrt{2} \sqrt{1 + \sqrt{3} i}$ represents a point P in the Argand plane and P lies in the third quadrant, then the polar form of z is

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$2 [\cos (\frac{- 4 π}{3}) + i \sin (\frac{- 4 π}{3})]$

$2 [\cos (\frac{- 5 π}{6}) + i \sin (\frac{- 5 π}{6})]$

$2 [\cos (\frac{- π}{6}) + i \sin (\frac{- π}{6})]$

$2 [\cos (\frac{- 2 π}{3}) + i \sin (\frac{- 2 π}{3})]$

answer is B.

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Detailed Solution

$z = \sqrt{2} \sqrt{1 + \sqrt{3} i} = 2 \sqrt{\frac{1}{2} + i \frac{\sqrt{3}}{2}}$

$z = 2 {(\cos \frac{π}{3} + i \sin \frac{π}{3})}^{\frac{1}{2}} = 2 (\cos \frac{2 kπ + \frac{π}{3}}{2} + i \sin \frac{2 k π + \frac{π}{3}}{2}), k = 0, 1 = 2 c i s \frac{π}{6}, 2 c i s \frac{7 π}{6} o r 2 c i s \frac{- 5 π}{6}$

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