If $\alpha \ne \beta$and ${\alpha }^2=5\alpha -3,{\beta }^2=5\beta -3$, then the equation having $\alpha /\beta$and $\beta /\alpha$ as its roots, is

We have

${\alpha }^2=5\alpha -3\Rightarrow {\alpha }^2-5\alpha +3=0\Rightarrow \alpha =\frac{5\pm \sqrt{13}}{2}$

Similarly, ${\beta }^2=5\beta -3\Rightarrow \beta =\frac{5\pm \sqrt{13}}{2}$

Since, $\alpha \ne \beta$

$\therefore \alpha =\frac{5+\sqrt{13}}{2}$ and $\beta =\frac{5-\sqrt{13}}{2}$

or, $\alpha =\frac{5-\sqrt{13}}{2}$ and $\beta =\frac{5+\sqrt{13}}{2}$

Thus, in either case, we obtain

${\alpha }^2+{\beta }^2=\frac{1}{4}(50+26)=19$,

and, $\alpha \beta =\frac{1}{4}(25-13)=3$, in both the cases.

Thus, the equation having $a/\beta and \beta / a$ as its roots is

$x^2-x\left(\frac{\alpha }{\beta }+\frac{\beta }{\alpha }\right)+\frac{\alpha \beta }{\alpha \beta }=0$

$\Rightarrow x^2-x\left(\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }\right)+1=0\Rightarrow 3x^2-19x+3=0$