Question

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

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

Answer 1

Answer: (B)

${\alpha }^{-2},{\beta }^{-2}$

$\alpha +\beta =\frac{b}{a},\alpha \beta =\frac{c}{a}$
Let $\gamma$ and $\delta$ be the roots of equation $c^2{y}^2+(2ca+b^2)y+a^2=0$
$\gamma +\delta =\frac{-(2ca+b^2)}{c^2},\gamma \delta =\frac{a^2}{c^2}$
So clearly we can see that $\gamma ={\alpha }^{-2},\delta ={\beta }^{-2}$
Hence, option 'B' is correct.