Complex Numbers

Given $sz + \overline{tz} + r = 0$...(1)On taking conjugate $\overline{s}\overline{z} + \overline{tz} + \overline{r} = 0$......(2)From (1) and (2) eliminating $\overline{z}$$z(|s|^2 - |t|^2) = \overline{r}t - \overline{rs}$A) If $|s| \neq |t|$ then z has unique valueB) If $|s| = |t|$ then $\overline{r}t - \overline{rs}$ may or may not be zero so L may be empty setC) locus of z is null set or singleton set or a line in all cases it will intersect given circle at most two pointsD) In this case locus of z is a line so L has infinite elements