Progression (Sequence and Series)Hard

Question

If a, b, c are in GP, then the equations ax2 + 2bx + c = 0 and dx2 + 2bx + f = 0 have a common root, if, are in

Options

A.AP
B.GP
C.HP
D.None of these

Solution

Since a, b, c are in GP.
⇒         b2 = ac
Given,     ax2 + abx + c = 0
⇒         ax2 + 2 x + c = 0
⇒         (√a x + √c)2 = 0
⇒        
Since, ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have common root.
∴       must satisfy dx2 + 2ex + f = 0
⇒         + f = 0
⇒         = 0
⇒              (∵b2 = ac)
∴         are in AP.

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