The determinant ∣∣
∣∣abaα+bbcbα+caα+bbα+c0∣∣
∣∣=0, if
A
a,b,c are in A.P.
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B
a,b,c are in G.P.
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C
a,b,c are in H.P.
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D
α is a root of the equation ax2+bx+c=0
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E
(x−α) is a factor of ax2+2bx+c
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Solution
The correct options are Ba,b,c are in G.P. E(x−α) is a factor of ax2+2bx+c
The given determinant is ∣∣
∣∣abaα+bbcbα+caα+bbα+c0∣∣
∣∣=0
Expanding by C3, =(aα+b)(b2α+bc−acα−bc)−(bα+c)(abα+ac−baα−b2) =(aα+b)(b2α−acα)−(bα+c)(ac−b2) =(b2−ac)(aα2+αb+bα+c) The determinant will be 0 if (b2−ac)=0 which is a geometric mean, or (aα2+2bα+c)=0 i.e, if (x−α) is the factor.