Question

If the sum of the roots of the quadratic equation $(b–c)x^2+(c–a)x+2(a–b)=0$ is equal to product of the roots, then the required condition is (where $a,b$ and $c$ are distinct positive real numbers)

• A. $a+b+c=0$
• B. $2a+2b+c=0$
• C. $a,b,c$ are in AP
• D. $a,b,c$ are in GP

Answer 1

The correct option is C $a,b,c$ are in AP

The given quadratic equation is $(b-c)x^2+(c-a)x+2(a-b)=0$
Let $\alpha$ and $\beta$ be the roots then,
$\alpha +\beta =\frac{-(c-a)}{b-c}=\frac{a-c}{b-c}$ and $\alpha \beta =\frac{2(a-b)}{(b-c)}$
According to the question:
$\alpha +\beta =\alpha \beta$
$\frac{(a-c)}{(b-c)}=\frac{2(a-b)}{(b-c)}$
$\Rightarrow a-c=2a-2b$
$\Rightarrow 2b=a+c$
$\therefore a,b,c$ are in A.P.
Hence, the correct answer is option (3).