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 $C=a^{2}c+c^{2}a-c{b}^2-c^{2}b$, where $a>b>c>0$ and the equation $A{x}^2+Bx+C=0$ has equal roots, then $a,b,c$ are

• A. NOT in A.P/G.P./H.P
• B. in A.P.
• C. in G.P.
• D. in H.P.

Answer 1

Answer: (D)

Given, $A=a^{2}b+a{b}^2-a^{2}c-a{c}^2=a^2(b-c)+a\left(b^2-c^2\right)=a(b-c)(a+b+c)$
Similarly, $B=b(c-a)(a+b+c)$ and $C=c(a-b)(a+b+c)$
Now, $A{x}^2+Bx+C=0$
$\Rightarrow (a+b+c)\left[a(b-c)x^2+b(c-a)x+c(a-b)\right]=0$ has equal roots. (Given)
Clearly, $x=1$ is one root of equation ( 1 , so other root is also 1 .
$\Rightarrow$ Product of roots $=1$ of equation $(1)\Rightarrow \frac{c(a-b)}{a(b-c)}=1\Rightarrow b=\frac{2ac}{a+c}$
$\therefore a,b,c$ are in H.P.