= Solution
A <projective conic> in $\mathbb P^2(\mathbb C)$ is the zero locus of a nonzero homogeneous quadratic <polynomial>, up to nonzero scaling of that <polynomial>. Under an invertible linear change of homogeneous coordinates, its <symmetric matrix> has rank $3$, $2$, or $1$. Over $\mathbb C$, the corresponding forms are a nonsingular conic such as $XZ-Y^2=0$, two distinct lines such as $XY=0$, or a double line $X^2=0$. In the last case the repeated factor records multiplicity, even though the underlying point set is one line.
A <projective line> is the projectivization of a two-dimensional <linear subspace> of $\mathbb C^3$, equivalently the zero locus of one nonzero linear form. For two such subspaces $U,W$, the <dimension formula for a sum of subspaces> gives $\dim(U\cap W)\geq2+2-3=1$. A nonzero vector in the intersection therefore supplies a common projective point. Distinct lines have exactly one such point.
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