Let s, t, r be non-zero complex numbers and L be the set of...

Given
$sz + t\overset{-}{z} + r = 0$ ...(i)
On taking conjugate $\overset{-}{s}\overset{-}{z} + \overset{-}{t}z + \overset{-}{r} = 0$ ...(ii)
$z$\left({\left|s\right|}^{2} - {\left|t\right|}^{2}\right) = \overset{-}{r}t - r\overset{-}{s}$<br />\left(1\right)$ If \left|s\right| \ne \left|t\right| then $z$ has unique value
\left(2\right) If \left|s\right| = \left|t\right| then $\overset{-}{r}t - r\overset{-}{s}$ may or may not be zero so $L$may be empty set
\left(3\right) Locus$of zis null set or$singleton set or a line in all cases it will intersect given circle at most two points.
\left(4\right) In this case locus of$z$ is a line so $L$ has infinite elements