Question

If $z_1{z}_2$ be two distinct complex numbers and let z = (1 - t) $z_1$ + t$z_2$ for some real number t with 0 < t < 1. If arg $\left(\omega \right)$ denotes the principal argument of a non-zero complex number $\left(\omega \right)$, then

• A. $|z-z_1|+|z-z_2|=|z_2-z_1|$
• B. $arg(z-z_1)=arg(z-z_2)$
• C. $\left|\begin{array}{cc}z-z_1& \overline{z}-\overline{z_1}\\ z_2-z_1& \overline{z_2}-\overline{z_1}\end{array}\right|$
• D. $arg(z-z_1)=arg(z_2-z_1)$

Answer 1

Answer: (A)

$|z-z_1|+|z-z_2|=|z_2-z_1|$

Given equation: $z=(1-t)z_1+t{z}_2$

So we can write, $|z-z_1|=t|z_2-z_1|=t|z_1-z_2|$ -------(1)

$z-z_2=(1-t)(z_1-z_2)$

$|z-z_2|=(1-t)|(z_1-z_2)|=(1-t)|z_2-z_1|$ ------(2)

From (1) and (2) we can say,

$|z-z_1|+|z-z_2|=|z_1-z_2|$