Question

If both the roots of $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal, then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. None of these

Answer 1

The correct option is C H.P.

$a(b-c)x^2+b(c-a)x+c(a-b)=0$
By observation, sum of coefficient $=0$, so $x=1$ is one root of the equation,
So both of the roots are $1$,
Product of roots
$\frac{c(a-b)}{a(b-c)}=1\Rightarrow ac-bc=ab-ac\Rightarrow 2ac=ab+bc\Rightarrow b=\frac{2ac}{a+c}$
Therefore,
$a,b,c$ are in H.P.