Question

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

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

Answer 1

The correct option is C HP

The roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ are equal

Since, roots of the equation are equal.

$\therefore$ Discriminant $=0$

i.e., $b^2{(c-a)}^2-4ac(b-c)(a-b)=0$

$\Rightarrow b^2{c}^2-2b^{2}ca+b^2{a}^2-4ac(ab-b^2-ac+bc)$

$\Rightarrow b^2{(c+a)}^2-4abc(a+c)+4a^2{c}^2=0$

$\Rightarrow {(b(a+c)-2ac)}^2=0\Rightarrow b=\frac{2ac}{a+c}$

$\Rightarrow a,b,c$ are in HP.