Question

If ${\alpha }^2=5\alpha -3,{\beta }^2=5\beta -3$ then the value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$ (where $\alpha \ne \beta$) is-

• A. $\frac{19}{3}$
• B. $\frac{25}{3}$
• C. $\frac{-19}{3}$
• D. $Noneofthese$

Answer 1

The correct option is A $\frac{19}{3}$

${\alpha }^2=5\alpha -3⟹{\alpha }^2-5\alpha +3=0$

${\beta }^2=5\beta -3⟹{\beta }^2-5\beta +3=0$

So, if we consider $\alpha ,\beta$ to be roots of the equation

$x^2-5x+3=0$

Now, $\alpha +\beta =5$

$\alpha \beta =3$

${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha +\beta =25-6=19$

$\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{19}{3}$