Question

If $\alpha ,\beta$ are the roots of quadratic equation $a{x}^2+bx+c=0$, then ${(a\alpha +b)}^{-2}+{(a\beta +b)}^{-2}=$

• A. $\frac{b^2-ac}{a^3{c}^2}$
• B. $\frac{b^2-2ac}{a^2{c}^2}$
• C. $\frac{b^3-3ac}{a^2{c}^2}$
• D. None of these

Answer 1

Answer: (B)

$\frac{b^2-2ac}{a^2{c}^2}$

Given $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$

$\therefore a{\alpha }^2+b\alpha +c=0\Rightarrow a\alpha +b=-\frac{c}{\alpha }$

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

$\therefore {(a\alpha +b)}^{-2}+{(a\beta +b)}^{-2}$

$=\frac{{\alpha }^2+{\beta }^2}{c^2}=\frac{1}{c^2}[{(\alpha +\beta )}^2-2\alpha \beta ]=\frac{b^2-2ac}{a^2{c}^2}$