Question

If z=x+iy is a complex number satisfying $|z+\frac{i}{2}|=|z-\frac{i}{2}|$, then the locus of 'z' is ...

• A. Real Axis
• B. Imaginary Axis
• C. y = x
• D. 2y = x

Answer 1

Answer: (A)

Real Axis

Given,
${|z+\frac{i}{2}|}^2={|z-\frac{i}{2}|}^2$

$\Rightarrow |z+\frac{i}{2}|=|z-\frac{i}{2}|$

Visualising in complex plane we can easing see that z lines in perpendiculer bisector of line seguent joining $\frac{i}{2}$ with $\frac{i}{2}$
i.e. the x-axis or y=0
theoreticelly,

${|z+\frac{i}{2}|}^2={|z-\frac{i}{2}|}^2$

$\Rightarrow {\left|\right(x)+i(y+1/2\left)\right|}^2={|x+i(y-1/2\left)\right|}^2$

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

$\Rightarrow y=-y$
or 2y = 0
or y = 0