Question

If $|a|=|b|\ne 0$ and $\overline{a}c=b\overline{c}$, then $az+b\overline{z}+c=0$ represents

• A. A circle
• B. An ellipse
• C. A straight line
• D. A point

Answer 1

The correct option is C A straight line

Let $a=i$ and $b=1$ so that $|a|=|b|$.
$\overline{a}=-i$ and $\overline{b}=1$.
Let $c=p+iq$
Thus, $-ic=\overline{c}=>-ip+q=p-iq=>-i(p-q)=(p-q)=>p-q=0=>p=q$.
Let $z=m+in$
$az+b\overline{z}+c=0=>iz+\overline{z}+p+iq=0=>im-n+m-in+p+iq=0=>(m-n+p)+i(m-n+q)=0$
Thus, $m-n+p=m-n+q=0=>p=q$, which is true. Thus, there is no contradiction. Also, it represents a straight line.
Hence, (C) is correct.