If a, b and c are in GP, then the equation ax^2 + 2bx + c = 0 and dx^2 + 2ex + f = 0 have a common root, if d / a, e / b and f / c are in

1) AP

2) HP

3) GP

4) None of these

Solution: (1) AP

Since a, b, c are in GP, b² = ac

ax² + 2bx + c = 0

ax² + 2 √ac x + c = 0

(√ax + √c)² = 0

x = – √c / a

Since ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have common root.

x = – √c / a must satisfy dx² + 2ex + f = 0

d * (c / a) – 2e (√c / a) + f = 0

(d / a) – (2e / √ac) + (f / c) = 0

(2e / b) = (d / a) + (f / c)

(d / a), (e / b), (f / c) are in AP.

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If a, b and c are in GP, then the equation ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root, if d / a, e / b and f / c are in