Question

If a, b c are in HP then the expression $a(b-c)x^2$ + b(c-a)x + c(a-b)

• A. has real and distinct factors
• B. is a perfect square
• C. has no real factor
• D. None of these

Answer 1

The correct option is B

is a perfect square

As a(b-c)+b(c-a)+c(a-b)=0, x = 1 is a root of the corresponding equation. The other root of the equation $=\frac{c(a-b)}{a(b-c)}=1$ because, a, b, c in HP implies $\frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b},$ i.e., $\frac{a-b}{a}=\frac{b-c}{c}$
$\therefore x=1,$ 1 are the roots of the corresponding equation. So, $(x-1{)}^2$ is a factor.