Question

The locus of point $z$ satisfying $Re\left(\frac{1}{z}\right)=k$, where $k$ 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 $z=x+iy$, then
$Re\left(\frac{1}{z}\right)=k\Rightarrow Re\left(\frac{1}{x+iy}\right)=k\Rightarrow Re\left(\frac{x-iy}{x^2+y^2}\right)=k\Rightarrow \frac{x}{x^2+y^2}=k\Rightarrow x^2+y^2-\frac{x}{k}=0$

Hence, the locus is a circle.