Question
If $$a, b, c$$ are in H.P. then the expression $$a\left( {b - c} \right){x^2} + b\left( {c - a} \right)x + c\left( {a - b} \right)$$
A.
has real and distinct factors
B.
is a perfect square
C.
has no real factor
D.
None of these
Answer :
is a perfect square
Solution :
As $$a\left( {b - c} \right) + b\left( {c - a} \right) + c\left( {a - b} \right) = 0,x = 1$$ is a root of the corresponding equation. The other root of the equation
$$ = \frac{{c\left( {a - b} \right)}}{{a\left( {b - c} \right)}} = 1$$ because $$a, b, c$$ in H.P. implies $$\frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b},\,{\text{i}}{\text{.e}}{\text{., }}\frac{{a - b}}{a} = \frac{{b - c}}{c}$$
∴ $$x = 1, 1$$ are the roots of the corresponding equation. So, $${\left( {x - 1} \right)^2}$$ is a factor.