The pair of lines represented by  $3{\mathrm{ax}}^2+5\mathrm{xy}+\left({\mathrm{a}}^2-2\right){\mathrm{y}}^2=0$ are perpendicular to each other for

These lines are ${\perp }^{\mathrm{r}}\text{, if }\mathrm{a}+\mathrm{b}=0$

$\Rightarrow 3\mathrm{a}+\left({\mathrm{a}}^2-2\right)=0\Rightarrow {\mathrm{a}}^2+3\mathrm{a}-2=0$

This is quadratic equation in ‘a’ and ${\mathrm{B}}^2-4\mathrm{AC}>0$

$\therefore$ the roots of the above equation are real and distinct

$\therefore$ the lines are ${\perp }^{\mathrm{r}}$ to each other for two values of ‘a’