Relationship between Zeroes and Coefficients of a Polynomial
If α is a c...
Question
If α is a complex constant such that αz2+z+¯¯¯¯α=0 has a real root, then
A
α+¯¯¯¯α=1
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B
α+¯¯¯¯α=0
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C
α+¯¯¯¯α=−1
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D
The absolute value of the real root is 1
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Solution
The correct options are Aα+¯¯¯¯α=1 Cα+¯¯¯¯α=−1 D The absolute value of the real root is 1 Let z=c be a real root. Then, ac2+c+¯¯¯a=0 (1) Putting α=p+iq, we have (p+iq)c2+c+p−iq=0 ⇒pc2+c+p=0 and qc2−q=0⇒c=±1(∵q≠0) ∴(1)⇒α±1+¯¯¯¯α=0 Also, |c|=1