Question

If z = x + iy and |z 1 + 2i | = | z + 1 2i |,then the locus of z is

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

Answer 1

The correct option is C straight line

We know that $z=x+iy$

$\Rightarrow |z-1+2i|=|z+1-2i|,$ locus of $x=?$

$\Rightarrow |x+iy-1+2i|=|x+iy+1-2i|$

$\Rightarrow |(x-1)+i(y+2)|=|(x+1)+i(y-2)|$

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

$\Rightarrow x^2-2x+1+y^2+4y+4=x^2+2x+1+y^2-4y+4$

$\Rightarrow -2x+4y+5=2x-4y+5$

$\Rightarrow 4x=8y$

$\Rightarrow x=2y$

$\therefore x-2y=0$

$\therefore$ The locus of $z$ is a straight line.

Hence, the answer straight line.