Question

If $z_1$ lies on the circle $|z|=3$ and $x+iy=z_1+\frac{1}{z_1}$ then locus of $z$ is

• A. circle
• B. parabola
• C. ellipse
• D. hyperbola

Answer 1

The correct option is C ellipse

Let $z_1=3\left(\cos \theta +i\sin \theta \right)$

$x+iy=z_1+\frac{1}{z_1}x+iy=3\left(\cos \theta +i\sin \theta \right)+\frac{1}{3\left(\cos \theta +i\sin \theta \right)}=3\left(\cos \theta +i\sin \theta \right)+\frac{1}{3}\left(\cos \theta +i\sin \theta \right)x+iy=\frac{10}{3}\cos \theta +i\frac{8}{3}\sin \theta x=\frac{10}{3}\cos \theta y=\frac{8}{3}\sin \theta$

Therefore,

${\left(\frac{3x}{10}\right)}^2+{\left(\frac{3y}{8}\right)}^2=1\frac{x^2}{\frac{100}{9}}+\frac{y^2}{\frac{64}{9}}=1$

Ellipse

$\therefore$ Option $C$ is correct answer.