Question

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

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

Answer 1

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

Given: $a{x}^2-bx+c=0$ have roots $\alpha ,\beta$.
$\Rightarrow \alpha +\beta =\frac{b}{a}\text{and}\alpha \beta =\frac{c}{a}$
Take,$y=\frac{1}{x^2}$
$\Rightarrow a.\frac{1}{y}-\frac{b}{\sqrt{y}}+c=0$
$\Rightarrow a-b\sqrt{y}+cy=0$
$\Rightarrow b^{2}y=(a+cy{)}^2$

$\therefore (a+cy{)}^2=b^{2}y$
have roots as $\frac{1}{{\alpha }^2},\frac{1}{{\beta }^2}$