Question

Consider the equation $az+b\overline{z}+c=0$, where $a,b,cϵZ$

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

Answer 1

Given:
$az+b\overline{z}+c=0$, where $a,b,cϵZ$

$az+b\overline{z}+c=0\dots \left(i\right)$

Taking conjugate of (i), we get

$\overline{a}\overline{z}+\overline{b}z+\overline{c}=0\dots \left(ii\right)$

Adding both (i) and (ii)

$(a+\overline{b})z+(b+\overline{a})\overline{z}+(c+\overline{c})=0$

$Az+\overline{A}\overline{z}+B=0$, where $B=c+\overline{c}$ , is real

Hence, locus of $z$ is a straight line.