Question

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

• A. a straight line
• B. a circle
• C. an ellipse
• D. a hyperbola
• E. none of these

Answer 1

Answer: (B)

Let $z=x+iy,$ then $\mathrm{Re}\left(\frac{1}{z}\right)=k$
$\Rightarrow$ $\mathrm{Re}\left(\frac{1}{x+iy}\right)=k$
$\Rightarrow$ $\mathrm{Re}\left(\frac{x}{x^2+y^2}-\frac{iy}{x^2+y^2}\right)=k$
$\Rightarrow$ $k=\frac{x}{x^2+y^2}$
$\Rightarrow$ $x^2+y^2-\frac{1}{k}x=0$
which is an equation of circle.