Let be non-zero complex numbers and be the set of solutions of the equation

MATHEMATICS · Single correct · CBSE · Mathematics · Complex Numbers and Quadratic Equations · Complex Numbers and Quadratic Equations

Problem

Let S,t,LS, t, L be non-zero complex numbers and LL be the set of solutions z=x+iy,(x,y∈R,i=−1)z = x + iy, (x, y \in \mathbb{R}, i = \sqrt{-1}) of the equation sz+tzˉ+r=0sz + t\bar{z} + r = 0 , where zˉ=x−iy\bar{z} = x - iy . Then, which of the following statement(s) is(are) TRUE? (A) If LL has exactly one element, then ∣s∣≠∣t∣|s| \neq |t| . (B) If ∣s∣=∣t∣|s| = |t| , then LL has infinitely many elements. (C) The number of elements in L∩{z∣∣z−1+i∣=5}L \cap \{z \mid |z - 1 + i| = 5\} is at most 2. (D) If LL has more than one element, then LL has infinitely many solutions elements.

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