Is it possible to cover all circle area with infinite lines starting from the center?

Question

Is it possible to cover all area of a circle of radius r>0 with infinite lines starting from the center?

Answer 1

Yes, it is (assuming you mean "start from the center", and you mean "infinitely many lines"). For each point on the circle, take the line going through the origin and that point. Any point in the disc lies on one of these lines, so their union covers the entire disc (and in fact the entire plane).

Answer 2

Every point in the disk is on some radius, so the union of all the radii covers the disk.

There are infinitely many radii, and all are needed to complete the cover.

But note -- the set of radii is an uncountable set. In other words, you can't list them as a sequence.