JEE Mains · Maths · STD 11 - 4.1 complex nubers
If the equation \(a| z |^{2}+\overline{\bar{\alpha} z +\alpha \overline{ z }}+ d =0\) represents \(a\) circle where \(a,d\) are real constants then which of the following condition is correct ?
- A \(|\alpha|^{2}-a d \neq 0\)
- B \(|\alpha|^{2}-a d>0\) and \(a \in R-\{0\}\)
- C \(|\alpha|^{2}-a d \geq 0\) and \(a \in R\)
- D \(\alpha=0, a , d \in R ^{+}\)
Answer & Solution
Correct Answer
(B) \(|\alpha|^{2}-a d>0\) and \(a \in R-\{0\}\)
Step-by-step Solution
Detailed explanation
\(az \overline{ z }+\alpha \overline{ z }+\bar{\alpha} z + d =0 \rightarrow\) Circle centre \(=\frac{-\alpha}{ a } \quad 2=\sqrt{\frac{\alpha \bar{\alpha}}{ a ^{2}}-\frac{ d }{ a }}=\sqrt{\frac{\alpha \bar{\alpha}- ad }{ a ^{2}}}\) So \(|\alpha|^{2}- ad >0 \& a \in R -\{0\}\)
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