Question

Locus of the point $z$ satisfying $Re\left(\frac{1}{z}\right)=k$, $k$ is a non-zero real number, is:

• A. a straight line
• B. a circle
• C. an ellipse
• D. a hyperbola

Answer 1

The correct option is B a circle

Let $z=r{e}^{i\theta }$

$\frac{1}{z}=\frac{1}{r{e}^{i\theta }}=\frac{e^{-i\theta }}{r}$

$=\frac{\cos \theta -i\sin \theta }{r}$

$\Rightarrow k=\frac{\cos \theta }{r}$

Now, substituting $r$ as $\sqrt{x^2+y^2}$ and $\cos \theta$ as $\frac{x}{\sqrt{x^2+y^2}}$, we get

$k=\frac{x}{x^2+y^2}$

$\Rightarrow k{x}^2+k{y}^2-x=0$, which is a circle.