Question

Let $s,t,r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y (x , y \in \mathbb R, i = \sqrt{-1})$ of the equation $sz + t \bar{z} + r = 0$, where $\bar{z} = x - iy$. Then, which of the following statement(s) is (are) TRUE?

• A. If $L$ has exactly one element, then $|s| \neq |t|$
• B. If $|s| = |t|$, then $L$ has infinitely many elements
• C. The number of elements in $L \cap \{z : | z -1 + i| = 5 \}$ is at most 2
• D. If $L$ has more than one element, then $L$ has infinitely many elements

Answer 1

Answer: (A), (C), (D)

Given
$sz + t \bar{z} + r = 0 $ .....(1)
on taking conjugate $\bar{s} \bar{z} + \bar{t} z + \bar{z} + \bar{r} = 0 $ ........(2)
from (1) and (2) elliminating $\bar{z}$
$z (|s|^2 - |t|^2) = \bar{r} t - r \bar{s}$
(A) If $|s| \neq |t|$ then z has unique value
(B) If $|s| = |t| $ then $\bar{r} t - r \bar{s}$ may or may not be zero so L may be empty set
(C) locus of z is noll set or singleton set or a line in all cases it will intersect given circle at most two points.
(D) In this case locus of z is a line so L has infinite elements