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Question

If a,b,c are in H.P., then the equation a(bc)x2+b(ca)x+c(ab)=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 get
abac+bcab+acbc=0

Hence, x=1 is a root.

1a,1b,1c are in A.P. As a,b,c are in H.P.
Hence,2b=1a+1cOr 1b1c=1a1b.

cbbc=baab.

a(cb)=c(ba)(i)

The standard quadratic equation is mx2+nx+k=0

and Product of roots = km

Here, Product of roots =c(ab)a(bc).

=c(ba)a(cb).

=1 from (i).

Hence, product of roots is 1.
Therefore, both roots are 1.

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