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Question

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 D da,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
dr22er+f=0
d(ca)2e(cb)+f=0
da2eb+fc=0
da+fc=2eb
da,eb,fc are in A.P.

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