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 $C = a ^2 c + c ^2 a - cb ^2- c ^2 b$, where $a > b > c >0$ and the equation $Ax ^2+ Bx + C =0$ 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, $A = a ^2 b + a b ^2- a ^2 c - ac ^2= a ^2( b - c )+ a \left( b ^2- c ^2\right)= a ( b - c )( a + b + c )$
Similarly, $B = b ( c - a )( a + b + c )$ and $C = c ( a - b )( a + b + c )$
Now, $Ax ^2+ Bx + C =0$
$\Rightarrow ( a + b + c )\left[ a ( b - c ) x ^2+ b ( c - a ) x + c ( a - b )\right]=0$ 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 $(1) \Rightarrow \frac{c(a-b)}{a(b-c)}=1 \Rightarrow b=\frac{2 a c}{a+c}$
$\therefore a, b, c$ are in H.P.