Question

If $Re\left(\frac{1}{z}\right)>\frac{1}{2}$ and $Re\left(z\right)>0$, then which of the following is/are true about the locus of $z$?

• A. The locus of $z$ is the region inside a circle.
• B. The locus of $z$ is the region inside an ellipse.
• C. Area of locus is $\pi \text{sq. units}$
• D. Centre of the locus is $(1,0)$

Answer 1

Answer: (A), (D)

The correct options are
A The locus of $z$ is the region inside a circle.
D Centre of the locus is $(1,0)$

Let $z=x+iy$
$\frac{1}{z}=\frac{1}{x+iy}\Rightarrow \frac{1}{z}=\frac{x-iy}{x^2+y^2}$

As $Re\left(\frac{1}{z}\right)>\frac{1}{2}$, so
$\frac{x}{x^2+y^2}>\frac{1}{2}\Rightarrow 2x>x^2+y^2\Rightarrow x^2+y^2-2x<0\Rightarrow (x-1{)}^2+y^2<1$

It's the interior region of the circle with centre $(1,0)$ and the radius is $1\text{unit}$
Area is less than $\pi \text{sq. units}$