Question

Of $\alpha ,\beta$ are the roots of $a{x}^2+c=bx,$ then the equation $(a+cy{)}^2=b^{2}y$ in y has the roots

• A. ${\alpha }^{-1},{\beta }^{-1}$
• B. ${\alpha }^2,{\beta }^2$
• C. $\alpha {\beta }^{-1},{\alpha }^{-1}\beta$
• D. ${\alpha }^{-2},{\beta }^{-2}$

Answer 1

The correct option is D ${\alpha }^{-2},{\beta }^{-2}$

$a{x}^2-bx+c=0\alpha +\beta =\frac{b}{a},\alpha \beta =\frac{c}{a}Also,(a+cy{)}^2=b^{2}y\Rightarrow c^2{y}^2-(b^2-2ac)y+a^2=0or{\left(\frac{c}{a}\right)}^2{y}^2-({\left(\frac{b}{a}\right)}^2-2\left(\frac{c}{a}\right))y+1=0or(\alpha \beta {)}^2{y}^2-({\alpha }^2+{\beta }^2)y+1=0or{y}^2-({\alpha }^{-2}+{\beta }^{-2})y+{\alpha }^{-2}{\beta }^{-2}=0or(y-{\alpha }^{-2})(y-{\beta }^{-2})=0$
$\therefore$ Roots are ${\alpha }^{-2},{\beta }^{-2}$