If `a,b,c` are in GP, show that the equations `ax^(2)+2bx+c=0` and `dx^(2)+2ex+f=0` have a common root if `a/d,b/e,c/f` are in HP

Options

A. ` ( d ) /(a), ( e ) /(b) , ( f)/(c ) ` are in GP
B. ` d, e, f ` are in GP
C. ` d, e, f ` are in AP
D. ` (d ) / (a), (e )/(b), (f ) / (c ) ` are in GP

Explanation

` b ^ 2 = ac `
roots of ` ax ^ 2 + 2bx + c = 0 ` are equal i.e ` - (b ) /(a) `
` d ( - (b)/(a))^ 2 + 2 e ( - (b) /(a)) + f = 0 `
` db ^ 2 - 2bea + f a ^ 2 = 0 , dc - 2eb + fa= 0 `
divide by ac
` (dc ) /(ac ) - (2eb ) /(b ^ 2 ) + ( fa ) /(ac) = 0 rArr (d ) /(a) - ( 2eb) /(b ^ 2 ) + (fa ) /(ac) = 0 rArr (d ) /(a) - (2e)/(b ) + (f )/(c) = 0 `
` (d ) /(a), (e ) /(b), (f ) /(c) ` are in AP