Question

If the Imaginary part of $\frac{(2z+1)}{(iz+1)}$ is -2, then the locus of the point representing z in the complex plane is:

• A. A circle
• B. A straight line
• C. A parabola
• D. None of these

Answer 1

The correct option is B

A straight line

$\frac{2z+1}{iz+1}$ = $\frac{(2x+1)+2iy}{(1-y)+i^2{x}^2}$

= $\frac{\left[\right(2x+1)+2iy].\left[\right(1-y)-ix]}{(1-y{)}^2+i^2{x}^2}$

= $\frac{(2x-y+1)-(2x^2+2y^2+x-2y)i}{1+x^2+y^2-2y}$

$\therefore$ Imaginary part

$\frac{(2x^2+2y^2+x-2y)}{1+x^2+y^2-2y}$ = -2

$\Rightarrow x+2y-2$ = 0, Which is a straight line.