Question

If $|z-z_1|=|z-z_2|$ then the locus of $z$ is

• A. A circle with centre at the origin
• B. A circle with centre at $z_1$
• C. A straight line passing through the origin
• D. Is a perpendicular bisector of the line joining $z_1$ and $z_2$

Answer 1

The correct option is D Is a perpendicular bisector of the line joining $z_1$ and $z_2$

Let $z=x+iy$
Then
$|z-z_1|=|z-z_2|$
$|x+iy-(x_1+i{y}_1)|=|x+iy-(x_2+i{y}_2)|$
$|(x-x_1)+i(y-y_1)|=|(x-x_2)+i(y-y_2)|$
$|(x-x_1)+i(y-y_1){|}^2=|(x-x_2)+i(y-y_2){|}^2$
$(x-x_1{)}^2+(y-y_1{)}^2=(x-x_2{)}^2+(y-y_2{)}^2$
Therefore, the point $(x,y)$ is equidistant from the points $(x_1,y_1)$ and $(x_2,y_2)$. Therefore, the point $(x,y)$ will lie on the perpendicular bisector of the line joining $(x_1,y_1)$ and $(x_2,y_2)$.