If three distinct numbers a,b,c are in G.P. and the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root, then which one of the following statements is correct ?
A
d,e,f are in G.P.
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B
d,e,f are in A.P.
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C
da,eb,fc are in G.P.
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D
da,eb,fc are in A.P.
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Solution
The correct option is Dda,eb,fc are in A.P. a,b,c are in G.P. Let a=A,b=Ar,c=Ar2 ax2+2bx+c=0 ⇒Ax2+2Arx+Ar2=0 ⇒A(x+r)2=0 ⇒x=−r
x=−r is also a root of dx2+2ex+f=0 ∴dr2−2er+f=0 ⇒d(ca)−2e(cb)+f=0 ⇒da−2eb+fc=0 ⇒da+fc=2eb ⇒da,eb,fc are in A.P.