Question

When $\frac{z+i}{z+2}$ is purely imaginary then z lies on

• A. Circle of radius
• B. circle of radius
• C. straight line
• D. parabola

Answer 1

The correct option is A

Circle of radius

Let z = x + iy
$\frac{z+i}{z+2}$ = $\frac{(x+iy)+i}{(x+iy)+2}$
$\frac{x+i(y+1)}{(x+2)+iy}\times \frac{(x+2)-iy}{(x+2)-iy}$
This is purely imaginary. So the real part will be zero. So consider only real part.
$\Rightarrow$(x + 2)x + (y + 1)y = 0
$x^2$ + 2x + $y^2$ + y = 0
$x^2$ + 2x + $y^2$ + y = 0
$(x+1{)}^2$ + $(y+{\frac{1}{2}}^2)$ = 1 + $\frac{1}{4}$
= $\frac{5}{4}$
Which is a circle of radius $\sqrt{\frac{5}{4}}$ or $\sqrt{\frac{5}{2}}$.