Question

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

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

The correct option is C HP

Since, a(b – c) + b (c – a) + c (a – b)= 0 $\therefore x=1$ is a root of the given equation. Since, both theroots are equal,
therefore the other root is also 1.
Their sum = 1 + 1 = 2
$\Rightarrow \frac{-b(c-a)}{a(b-c)}=2\Rightarrow -bc+ab=2ab=2ac$
$\Rightarrow 2ac=b(a+c)\therefore b=\frac{2ac}{a+c}$
Hence, a, b, c are in HP.