 Solution

Assume $L\ne 0$. The quadratics divisible by $L$ form $L\cdot \mathrm{span}\{x,y\}$, a two-dimensional linear subspace; their projectivization is a projective line. Evaluating at the projective zero $[-\beta :\alpha ]$ of $L$ gives its equationAt the point $[L^2]=[{\alpha }^2:2\alpha \beta :{\beta }^2]$ of $C$, the tangent equation obtained from the gradient of $b^2-4ac$ is $-4{\beta }^{2}a+4\alpha \beta b-4{\alpha }^{2}c=0$. It is exactly the same line, establishing tangency.