If three distinct numbers a , b , c are in G.P. and the equations a x 2 + 2 b x + c = 0 and d x 2 + 2 e x + f = 0 have a common root, then which one of the following statements is correct?

Step-by-step solution

∵   a ,   b ,   c are in G.P. ⇒ b 2 = a c … ( i ) Equation a x 2 + 2 b x + c = 0 Hence, D = 4 b 2 - 4 a c = 0 ∵ b 2 = a c ⇒ equation has equal roots ⇒ roots are = - b a ,   - b a ( ∵ sum of roots α + α = - 2 b a ) ∵ Both equations have one root in common. ∴ Common root is - b a . It will satisfy equation. ∴ d - b a 2 + 2 e - b a + f = 0 ⇒ d b 2 - 2 a e b + a 2 f = 0 ⇒ d a c - 2 a e b + a 2 f = 0 ∵ b 2 = a