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The determinant 

Δ=abaα+bbcbα+caα+bbα+c0 equals zero, if 

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a
a, b, c are in A.P.
b
a, b, c are in H.P.
c
x-α is a factor of  ax2+2bx+c
d
x-α is a factor of ax2+bx+c

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detailed solution

Correct option is C

Applying , C3→C3-αC1-C2 we get , =-aα2+2bα+cabbc   [expanding along C3=-aα2+2bα+cac-b2= 0 if x-α is a factor of  ax2+2bx+c


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