Question

If $\alpha$ and $\beta$are the roots of the equation $a{x}^2+bx+c=0,$ then the equation, whose roots are $2\alpha +3\beta$ and $3\alpha +2\beta$, is

• A. $ac{x}^2+(a+c)bx-{(a+c)}^2=0$
• B. $ac{x}^2–(a+c)bx-{(a+c)}^2=0$
• C. $ab{x}^2–(a+b)cx-{(a+c)}^2=0$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

Find the equation :

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

$\Rightarrow \alpha +\beta =–b/a,\alpha \beta =c/a$

The required equation is

$$\begin{gathered} x^2–(2\alpha +3\beta +3\alpha +2\beta )+(2\alpha +3\beta )(3\alpha +2\beta )=0 \\ \Rightarrow x^2–(5\alpha +5\beta )x+(6{\alpha }^2+6{\beta }^2+13\alpha \beta )=0 \\ \Rightarrow x^2–5(\alpha +\beta )x+6{(\alpha +\beta )}^2+\alpha \beta =0 \\ \Rightarrow x^2–5\left(\frac{-b}{a}\right)x+6{\left(\frac{-b}{a}\right)}^2+\left(\frac{c}{a}\right)=0 \\ \Rightarrow x^2+\left(\frac{5b}{a}\right)x+6\left(\frac{b^2}{a^2}\right)+\left(\frac{c}{a}\right)=0 \\ \Rightarrow a^2{x}^2+6b^2+ca=0 \end{gathered}$$

Hence, the correct option is D.