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)

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

Answer 2

Let $w=z+\frac{1}{z}​=4e^{iθ}+\frac{1}{4}​e^{−iθ}$

$⇒w=\frac{17}{4}​\;cos\;θ+i\frac{15}{4}​\;sin\;θ$

So locus of w is ellipse $\frac{x^2}{(\frac{17}{4})^2}+\frac{y^2}{(\frac{15}{4})^2}=1$

Locus of $z$ is circle $x^2+y^2=16$

So intersect at $4$ points

The Correct Option is (B): the curves $C_1$ and $C_2$ intersect at 4 points