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  • For all z ∈ C on the curve C1: |z| = 4, let the locus of the point  

For all z ∈ C on the curve C1: |z| = 4, let the locus of the point  z+\frac{1}{z}   be the curve  C_2. Then :

Option: 1

the curve  C_1  lies inside C_2 


Option: 2

the curve  C_2  lies inside   C_1


Option: 3

the curves  C_1   and  C_2  intersect at 4 points


Option: 4

the curves  C_1   and  C_2  intersect at 2 points


Answers (1)

best_answer

\begin{aligned} & C_1:|z|=4 \text { then } z \bar{z}=16 \\ & z+\frac{1}{z}=z+\frac{\bar{z}}{16} \\ & =x+i y+\frac{x-i y}{16} \\ & z+\frac{1}{z}=\frac{17 x}{16}+i \frac{15 y}{16} \end{aligned}

Let  X=\frac{17 x}{16}, \quad Y=\frac{15}{16} y

\begin{aligned} & \frac{\mathrm{X}}{\left(\frac{17}{16}\right)}=x, \frac{Y}{\left(\frac{15}{16}\right)}=y \\ & \because x^2+y^2=16 \\ & \frac{X^2}{\left(\frac{17}{16}\right)^2}+\frac{Y^2}{\left(\frac{15}{16}\right)^2}=16 \\ & \Rightarrow C_2: \frac{x^2}{\left(\frac{17}{4}\right)^2}+\frac{y^2}{\left(\frac{15}{4}\right)^2}=1 \end{aligned}               (Ellipse)

 

Curve C1 and C2 intersect at 4 point.

 

Posted by

Rishabh

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