Question

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

• A. the curve $C_1$ lies inside $C_2$
• B. the curves $C_1$ and $C_2$ intersect at 4 points
• C. the curve $C_2$ lies inside $C_1$
• D. the curves $C_1$ and $C_2$ intersect at 2 points

Answer 1

Answer: (B)

Let $w=z+\frac{1}{z}=4e^{i\theta }+\frac{1}{4}{e}^{-i\theta }$
$\Rightarrow w=\frac{17}{4}\cos \theta +i\frac{15}{4}\sin \theta$
So locus of w is ellipse $\frac{x^2}{{\left(\frac{17}{4}\right)}^2}+\frac{y^2}{{\left(\frac{15}{4}\right)}^2}=1$
Locus of $z$ is circle $x^2+y^2=16$
So intersect at 4 points