Constant length of line segment through $\alpha(t)$ and $\beta(t)$

Question

"Let $\alpha(t)$ and $\beta(t)$ be regular curves on the plane ($\mathbb{R}^2$) such that, for all $t$, the line through $\alpha(t)$ and $\beta(t)$ is orthogonal to $\alpha$ and $\beta$ in $t$. Prove that the line segment between $\alpha(t)$ and $\beta(t)$ has constant length (i.e., it's length is independent of $t$). "
Well, this seems pretty simple, but I am struggling a bit to equate the line, since the parameter $t$ appears a lot: let $r(t)$ be the line determined by the points $\alpha(t)$ and $\beta(t)$, in such a way that $r(t) = \alpha(t) + (\beta(t) - \alpha(t)) \, t$, for $t \in \mathbb{R}$.
Now, looking at $\ell(t) = \left\| \beta(t) - \alpha(t)\right\|^2 = \left\langle \beta(t) - \alpha(t), \, \beta(t) - \alpha(t)\right\rangle $ and differentiating, one has
$\ell'(t) = 2 \left\langle \beta'(t) - \alpha'(t), \, \beta(t) - \alpha(t)\right\rangle$; how can we conclude that this inner product is zero?