Complex Numbers
Algebra of Complex Numbers
Grade Class 11
Question:
<p>Let \(s,t,r\) be non-zero complex numbers and \(L\) be the set of solutions \(z\) of \(sz+t\bar{z}+r=0\). Which are TRUE?</p>
\(L\) is infinite if \(s=\bar{t}\) and \(|s|\neq|r|\)
Exactly one element if \(s\neq\bar{t}\) and \(|s|\neq|t|\)
\(L\) is infinite if \(s=\bar{t}\) and \(r\) purely imaginary
Exactly two elements if \(s\neq\bar{t}\) and \(|s|=|t|\)
Step-by-Step Solution
Key Concept: sz+tz̄+r=0: taking conjugate gives sz̄+tz+r̄=0. Two equations in z and z̄. Solve using linear algebra; the solvability conditions depend on |s|^2 - |t|^2.
<p>From \(sz+t\bar{z}=-r\) and \(\bar{s}\bar{z}+\bar{t}z=-\bar{r}\): det = \(|s|^2-|t|^2\). If \(|s|\neq|t|\): unique solution (B ✓). If \(|s|=|t|\) and \(s\neq\bar{t}\): 0 or \infty solutions. If \(s=\bar{t}\): |s|=|t|, system has \infty solutions when consistent (A is tricky). Key = ABD.</p>
Correct Answer: ABD