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Question

# If a,b,c are in H.P., then the equation a(bâˆ’c)x2+b(câˆ’a)x+c(aâˆ’b)=0

A
has real and distinct roots.
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B
has equal roots.
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C
has no real roots.
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D
has 1 as a root.
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Solution

## The correct options are B has equal roots. D has 1 as a root.Substituting x=1, we getab−ac+bc−ab+ac−bc=0Hence, x=1 is a root.1a,1b,1c are in A.P. As a,b,c are in H.P.Hence,2b=1a+1cOr 1b−1c=1a−1b.c−bbc=b−aab.a(c−b)=c(b−a)……(i)The standard quadratic equation is mx2+nx+k=0 and Product of roots = kmHere, Product of roots =c(a−b)a(b−c).=c(b−a)a(c−b).=1…… from (i).Hence, product of roots is 1.Therefore, both roots are 1.

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