Existence of Special Types Primitive Pairs in Finite Fields Avoiding Affine Hyperplanes

Let $\text{Fm}$ be finite fields of order $q^m$, where $m\ge 2$ and $q$, a prime power. Given $\text{F}$-affine hyperplanes $A_1,\dots ,A_m$ of $\text{Fm}$ in general position, we study the existence of primitive element $\alpha$ of $\text{Fm}$, such that $f(\alpha )$ is also primitive, where $a{x}^2+bx+c\in \text{Fm}[x]$ ($a\mathrm{\ne }0$ and $b^2\mathrm{\ne }4ac$) in $\text{Fm}$ and the primitive pair $(\alpha ,f(\alpha ))$ avoids each $A_i$. We establish results for fields of higher order.