Question

Let $z_1,z_2$ be two fixed complex numbers in the Argand plane and $z$ be an arbitary point satisfying $|z-z_1|+|z-z_2|=|z_1-z_2|$. Then the locus of $z$ will be

• A. an ellipse
• B. a straight line joining $z_1$ and $z_2$
• C. a parabola
• D. a bisector of the line joining $z_1$ and $z_2$

Answer 1

The correct option is B a straight line joining $z_1$ and $z_2$

$|z-z_1|$ shows the distance from a point on locus $z$ to $z_1$ ( $z_1$ is a fixed point).

Similarly, $|z-z_2|$ shows the distance between the same point on locus $z$ to $z_2$.

On adding both the distance it must be equal to $|z_1-z_2|$, which shows the distance between $z_1$ and $z_2$.

So we find that any such point always on the line joining $z_1$ and $z_2$.

Thus the locus will be a straight line joining $z_1$ and $z_2$.