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 $w = z +\frac{1}{ z }=4 e ^{ i \theta}+\frac{1}{4} e ^{-i \theta}$
$\Rightarrow w =\frac{17}{4} \cos \theta+ i \frac{15}{4} \sin \theta$
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