Question

Let $z_1$ and $z_2$ 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 $\text{Arg}\left(w\right)$ denotes the principal argument of a non-zero complex number $w$, then

• A. $\text{Arg}(z–z_1)=\text{Arg}(z_2–z_1)$
• B. $|z–z_1|+|z–z_2|=|z_1–z_2|$
• C. $\text{Arg}(z–z_1)=\text{Arg}(z–z_2)$
• D. $\left|\begin{array}{cc}z-z_1& \overline{z}-\overline{z_1}\\ z_2-z_1& \overline{z_2}-\overline{z_1}\end{array}\right|=0$

Answer 1

Answer: (A), (B), (D)

The correct options are
A $\text{Arg}(z–z_1)=\text{Arg}(z_2–z_1)$
B $|z–z_1|+|z–z_2|=|z_1–z_2|$
D $\left|\begin{array}{cc}z-z_1& \overline{z}-\overline{z_1}\\ z_2-z_1& \overline{z_2}-\overline{z_1}\end{array}\right|=0$


We have $z=(1-t)z_1+t{z}_2,0<t<1$,
$\Rightarrow z=\frac{t{z}_2+(1-t)z_1}{t+1-t}$
$\Rightarrow z=t{z}_2+(1-t)z_1$
$z_1,z_2,z_3$ are collinear
So $\text{Arg}(z-z_1)=\text{Arg}(z-z_2)=\text{Arg}(z_2-z_1)$