Question

If the imaginary part of $\frac{2z+1}{iz+1}$ is $-2$, then the locus of point represented by $z$ is a

• A. Parabola
• B. Circle
• C. Straight line
• D. None of these

Answer 1

Answer: (C)

Straight line

Let $z=x+iy$
$\therefore \frac{2z+1}{1+iz}=\frac{(2x+1)+2iy}{(1-y)+ix}$
$=\frac{(2x+1)+2iy}{(1-y)+ix}\times \frac{(1-y)-ix}{(1-y-ix)}$
$\Rightarrow \left\{\frac{(2x+1)(1-y)+2xy}{(1-y{)}^2+x^2}\right\}+i\left\{\frac{2y(1-y)-x(2x+1)}{(1-y{)}^2+x^2}\right\}$
On comparing imaginary parts
$2y(1-y)-x(2x+1)=-2\left[\right(1-y{)}^2+x^2]$
$\Rightarrow 2y-2y^2-2x^2-x=-2-2y^2+4y-2x^2$
$\Rightarrow 2y+x=2$
Which is a straight line
Hence (c) is correct answer.