  Let $r,n>1$ be integers and $q$ be any prime power $q$ such that $r\mid
q^n-1$. We say that the extension $\mathbb{F}_{q^n}/\mathbb{F}_q$ possesses the
line property for $r$-primitive elements property if, for every
$\alpha,\theta\in\mathbb{F}_{q^n}^*$, 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 $(q^n-1)/r$. We prove
that, for sufficiently large prime powers $q$, $\mathbb{F}_{q^n}/\mathbb{F}_q$
possesses the line property for $r$-primitive elements. We also discuss the
(weaker) translate property for extensions.
