The translate and line properties for 2-primitive elements in quadratic extensions

Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid q^n-1$\mathrm{.}Wesaythattheextension$$\mathbb{F}$_{q^n}/$\mathbb{F}$q$ possesses the line property for $r$-primitive elements if, for every $\alpha ,\theta \in {\mathbb{F}}_{q^n}^{\ast }$, such that ${\mathbb{F}}_{q^n}={\mathbb{F}}_q(\theta )$, there exists some $x\in {\mathbb{F}}_q$, such that $\alpha (\theta +x)$ has multiplicative order $(q^n-1)\mathrm{/}r$. Likewise, if, in the above definition, $\alpha$ is restricted to the value $1$, we say that ${\mathbb{F}}_{q^n}\mathrm{/}{\mathbb{F}}_q$ possesses the translate property. In this paper we take $r=n=2$ (so that necessarily $q$ is odd) and prove that ${\mathbb{F}}_{q^2}\mathrm{/}{\mathbb{F}}_q$ possesses the translate property for 2-primitive elements unless $q\in \{5,7,11,13,31,41\}$. With some additional theoretical and computational effort, we show also that $$\mathbb{F}${q^2} /$\mathbb{F}$_q$possessesthelinepropertyfor2-primitiveelementsunless$q \in {3,5,7,9,11,13,31,41}$.