Question

Suppose $A,B,C$ are defined as $A=a^{2}b+a{b}^2-a^{2}c-a{c}^2,$ $B=b^{2}c+b{c}^2-a^{2}b-a{b}^2$ and $C=a^{2}c+a{c}^2-b^{2}c-b{c}^2$ respectively, where $a>b>c>0$. If the equation $A{x}^2+Bx+C=0$ has equal roots, then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. A.G.P.

Answer 1

The correct option is C H.P.

$A=a(b-c)(a+b+c)$
$B=b(c-a)(a+b+c)$
$C=c(a-b)(a+b+c)$

$A{x}^2+Bx+C=0$
$\Rightarrow (a+b+c)\left[a\right(b-c)x^2+b(c-a)x+c(a-b\left)\right]=0$
$\Rightarrow a(b-c)x^2+b(c-a)x+c(a-b)=0[\because a+b+c>0]$
Now, $a(b-c)+b(c-a)+c(a-b)=0$
$\Rightarrow 1$ is a root of $A{x}^2+Bx+C=0$.
Given that roots are equal.
$\Rightarrow 1,1$ are the roots.
Then, product of roots $=\frac{c(a-b)}{a(b-c)}=1$
$\Rightarrow ca-bc=ab-ac$
$\Rightarrow 2ac=ab+bc$
$\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$
$\Rightarrow a,b,c$ are in H.P.