Complex Number Question 8
Question 8 - 31 January - Shift 1
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
(1) the curves $C_1$ and $C_2$ intersect at 4 points
(2) the curves $C_1$ lies inside $C_2$
(3) the curves $C_1$ and $C_2$ intersect at 2 points
(4) the curves $C _{\text{}}$ lies inside $C _{\text{, }}$
Show Answer
Answer: (1)
Solution:
Formula: Equation of Ellipse, Equation of Circle
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 $\frac{x^{2}}{(\frac{17}{4})^{2}}+\frac{y^{2}}{(\frac{15}{4})^{2}}=1$
Locus of $z$ is circle $x^{2}+y^{2}=16$
