Question

The curve represented by the equation $Re\left(\frac{1}{z}\right)=2$ is;

• A. Circle with centre $(\frac{1}{4},0)$
• B. Straight line through $(\frac{1}{2},0)$
• C. Ellipse
• D. None of these

Answer 1

Answer: (A)

Circle with centre $(\frac{1}{4},0)$

$Re\left(\frac{1}{z}\right)=2$
Let, $z=x+iy$
$\frac{1}{z}=\frac{1}{x+iy}\times \frac{x-iy}{x-iy}$
$\frac{1}{z}=\frac{x-iy}{x^2-\left(i^2{y}^2\right)}$
$\frac{1}{z}=\frac{x-iy}{x^2+y^2}$
$\frac{1}{z}=\frac{x}{x^2+y^2}-\frac{iy}{x^2+y^2}$
$Re\left(\frac{1}{z}\right)=\frac{x}{x^2+y^2}=2$
$x=2x^2+2y^2$
$x^2+y^2-\frac{x}{2}=0$
$x^2-\frac{x}{2}+{\left(\frac{1}{4}\right)}^2-{\left(\frac{1}{4}\right)}^2+y^2=0$
${(x-\frac{1}{4})}^2+y^2=\frac{1}{16}$
$\therefore$ Circle with centre $(\frac{1}{4},0)$