For all z in mathbb{C} on the curve C_1: |z| | vedclass

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(A) Let $z = 4e^{i	heta}$. Then $w = z + \frac{1}{z} = 4e^{i	heta} + \frac{1}{4}e^{-i	heta}$.$w = 4(\cos 	heta + i \sin 	heta) + \frac{1}{4}(\cos

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the curves $C_1$ and $C_2$ intersect at $4$ points

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(A) Let $z = 4e^{i	heta}$. Then $w = z + \frac{1}{z} = 4e^{i	heta} + \frac{1}{4}e^{-i	heta}$.$w = 4(\cos 	heta + i \sin 	heta) + \frac{1}{4}(\cos 	heta - i \sin 	heta) = \left(4 + \frac{1}{4}ight) \cos 	heta + i \left(4 - \frac{1}{4}ight) \sin 	heta$.$w = \frac{17}{4} \cos 	heta + i \frac{15}{4} \sin 	heta$.Let $w = x + iy$. Then $x = \frac{17}{4} \cos 	heta$ and $y = \frac{15}{4} \sin 	heta$.The locus of $w$ is the ellipse $\frac{x^2}{(17/4)^2} + \frac{y^2}{(15/4)^2} = 1$.The curve $C_1$ is a circle $x^2 + y^2 = 16$.Since the semi-major axis $a = 17/4 = 4.25 > 4$ and the semi-minor axis $b = 15/4 = 3.75 < 4$,the ellipse intersects the circle at $4$ points.

For all $z \in \mathbb{C}$ on the curve $C_1: |z| = 4$,let the locus of the point $w = z + \frac{1}{z}$ be the curve $C_2$. Then:

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