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Question

# Suppose A,B,C are defined as A=a2b+ab2−a2c−ac2, B=b2c+bc2−a2b−ab2 and C=a2c+ac2−b2c−bc2 respectively, 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.A=a(b−c)(a+b+c) B=b(c−a)(a+b+c) C=c(a−b)(a+b+c) Ax2+Bx+C=0 ⇒(a+b+c)[a(b−c)x2+b(c−a)x+c(a−b)]=0 ⇒a(b−c)x2+b(c−a)x+c(a−b)=0 [∵a+b+c>0] Now, a(b−c)+b(c−a)+c(a−b)=0 ⇒1 is a root of Ax2+Bx+C=0. Given that roots are equal. ⇒1,1 are the roots. Then, product of roots =c(a−b)a(b−c)=1 ⇒ca−bc=ab−ac ⇒2ac=ab+bc ⇒2b=1a+1c ⇒ a,b,c are in H.P.

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