JEE Advanced 2018 · Paper 2, Mathematics question 5

Pair with the conjugate. Take the equation together with its conjugate:
$$sz+t\bar z=-r,\qquad\bar tz+\bar s\bar z=-\bar r.$$
Eliminating $\bar z$ gives $(|s|^2-|t|^2)z=t\bar r-\bar sr$.

• If $|s|\ne|t|$, there is exactly one solution.
• If $|s|=|t|$, there is either no solution or a whole line of solutions.

Check the options.

• (A) True: exactly one solution forces $|s|\ne|t|$.
• (B) False: $|s|=|t|$ can also give no solution.
• (C) True: $L$ is a point or a line, and either meets a circle in at most 2 points.
• (D) True: more than one solution means $|s|=|t|$, and then $L$ is a whole line.

(A), (C), (D)