Question

If $\alpha and\beta$ are the roots of the quadratic equation $x^2–5x+2=0,$ then a quadratic equation whose roots are can be $\frac{\alpha }{\beta }and\frac{\beta }{\alpha }$ can be

• A. $2x^2–21x+2=0$
• B. $4x^2–19x+4=0$
• C. $x^2–21x+1=0$
• D. $x^2–19x+1=0$

Answer 1

The correct option is C $x^2–21x+1=0$

Given that $\alpha and\beta$ are the root of $x^2-5x+2=0.$
$\Rightarrow \alpha +\beta =\frac{-b}{a}=\mathrm{5.......}\left(i\right)$
and $\alpha \beta =\frac{c}{a}=\mathrm{2...........}\left(ii\right)$
Now, $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }$
$=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta }$
$=\frac{5^2-2\times 2}{2}$
$\frac{25-4}{2}=\frac{21}{2}.......\left(iii\right)$
The quadratic equation whose roots are $\frac{\alpha }{\beta }and\frac{\beta }{\alpha }is{x}^2(\frac{\alpha }{\beta }+\frac{\beta }{\alpha })x+\frac{\alpha }{\beta }\times \frac{\beta }{\alpha }=0.$
$\Rightarrow x^2-\frac{21}{2}x+1=0\left(From\right(iii)$
$\Rightarrow 2x^2-21x+2=0$
Hence, the correct answer is option (1).