Question

If $a$, $b$, $c$ are in $H.P.$, and $\alpha$, $\beta$ are the roots of the equation $bc{x}^2+2cax+ab=0$, then the value of $\alpha +\beta +\alpha \beta$ is-

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

The correct option is A $0$

$a,b,c$ are in H.P

$\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$

$\Rightarrow b=\frac{2ac}{a+c}$

Let $\alpha$ and $\beta$ be the roots of $bc{x}^2+2cax+ab=0$

$\Rightarrow \alpha +\beta =-\frac{2ca}{bc}=-\frac{2a}{b}$

$\alpha \beta =\frac{ab}{bc}=\frac{a}{c}$

$\alpha +\beta +\alpha \beta =-\frac{2a}{b}+\frac{a}{c}$

$=\frac{-2ac+ab}{bc}$

$=\frac{-2a}{b}+\frac{a}{c}$

$=\frac{-2a}{\frac{2ac}{a+c}}+\frac{a}{c}$ where $b=\frac{2ac}{a+c}$

$=\frac{-2a(a+c)}{2ac}+\frac{a}{c}$

$=\frac{-a-c}{c}+\frac{a}{c}$

$=-\frac{a}{c}-1+\frac{a}{c}=-1$