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Question

If a,b,c are in G.P.,then the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root if daebfc are in
(i) A. P.
(ii) G.P.
(iii) H.P.
(iv) None of these.
Another form :
If ax2+2bx+c=0 and px2+2qx+r=0 have a common root and apbqcr in A.P. then prove that p,q,r are in G.P.

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Solution

(i).
a,b,c in G.P. b2ac=0
Δ1=4(b2ac)=0 It has equal roots
α+α=2ba or α=ba
Since the two equations have common roots therefore
α=b/a must also be a root of 2nd equation.
d(ba)2+2e(ba)+f=0
or da.b2a+2eb(b2a)+fcc=0 (Note)
or da.b2a2eb(b2a)+fc(b2a)=0
b2=ac
or da2eb+fc=0 or da,eb,fc are in A.P.
Another form : Assume that p,q,r are in G.P., so that q2=pr which implies that roots of 2nd equation are equal and rest as above.

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