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Question

Let A=a2b+ab2a2cac2, B=b2c+bc2a2bab2 and C=a2c+ac2b2cbc2, where a>b>c>0. If the equation Ax2+Bx+C=0 has equal roots, then a,b,c are in

A
A.P.
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B
G.P.
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C
H.P.
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D
A.G.P.
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Solution

The correct option is C H.P.
Given : A=a2b+ab2a2cac2, B=b2c+bc2a2bab2 and C=a2c+ac2b2cbc2
By observation, we see
A+B+C=0
Now, the given equation is
Ax2+Bx+C=0
So, one root of the equation is 1, as both roots are equal so both root are equal to 1,
Product of roots
CA=1A=Ca2b+ab2a2cac2=a2c+ac2b2cbc2a[a(bc)+(b2c2)]=c[c(ab)+(a2b2)]a(bc)(a+b+c)=c(ab)(a+b+c)abac=acbcb(a+c)=2acb=2aca+c

Hence, a,b,c are in H.P.

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