Does a plane algebraic curve of degree d>1 with $\frac{1}{2}d(d-1)$ singularities exist?

Question

In case it exists, it must be reducible, because the maximum number of singularities in an irreducible curve is $\frac{1}{2}(d-1)(d-2)<\frac{1}{2}d(d-1)$. Does it exist? Could we find an example?

Answer 1

Hint 1: $\frac{1}{2}d(d-1) = {d \choose 2}$.

Hint 2: If we take $d$ general lines in the plane, in how many points do two of them meet?