Let $k$ be a not-necessarily algebraically closed field of characteristic zero. Let $X$ be a positive-dimensional projective variety over $k$. Let $x$ be a closed point on $X$. Does there exist a curve over $k$ on $X$ which contains this point?

Variety = geometrically integral quasi-projective $k$-scheme

Curve = $1$-dimensional variety.

What about the special case where $x$ is a $k$-rational point?

I can blow-up $X$ at $x$ and take the image in $X$ of an effective ample Cartier divisor via this blow-up and reason by induction. But I'm afraid this doesn't give me a geometrically connected curve passing through $x$.