If $\lambda \ne \mu$and ${\lambda }^2=5\lambda -3,{\mu }^2=5\mu -3$, then the equation whose roots are $\frac{\lambda }{\mu }$and $\frac{\mu }{\lambda }$is

${\lambda }^2=5\lambda -3\Rightarrow {\lambda }^2-5\lambda +3=0$

${\mu }^2=5\mu -3\Rightarrow {\mu }^2-5\mu +3=0$

$\therefore \lambda ,\mu$ are roots of $x^2-5x+3=0$

$\therefore \lambda +\mu =5,\lambda \mu =3$

Then quadratic equation whose roots are

$\frac{\lambda }{\mu }\&\frac{\mu }{\lambda }$is $x^2-x\left(\frac{{\lambda }^2+{\mu }^2}{\lambda \mu }\right)+1=0$

$\Rightarrow 3x^2-19x+3=0$