I will be working in ZFC, but I am *not* assuming the Continuum Hypothesis (or Martin's Axiom). I know that it is consistent with ZFC that one can cover the real line with less than continuum of meager sets. My questions are about coverings by sets which are "nicer" than just meager (for instance, their proper intersections are finite). By "plane" below I mean the Euclidean plane and by "line" I mean an affine line. Of course, by Baire's theorem, Questions 1 and 2 both have negative answer if the sets $A$ are countable.

Question 1. Let $A$ be a set so that ${\aleph }_0=|{\mathbb N}|<|A| < 2^{{\aleph }_0} =|{\mathbb R}|$. Can one cover the plane by a family of lines $L_\alpha$, $\alpha\in A$?

In case I am missing some simple geometric consideration,

Question 2. For $A$ as above, can one cover the plane by a family of (real, irreducible) algebraic curves $X_\alpha$, $\alpha\in A$?

It feels as if these questions are related to the list compiled in this post by Joel David Hamkins, but I cannot quite figure out how.