= Solution
Assume $L\ne0$. The quadratics divisible by $L$ form $L\cdot\operatorname{span}\{x,y\}$, a two-dimensional <linear subspace>; their projectivization is a <projective line>. Evaluating at the projective zero $[-\beta:\alpha]$ of $L$ gives its equation
$$
\boxed{\beta^2a-\alpha\beta b+\alpha^2c=0}.
$$
At 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 $-4\beta^2a+4\alpha\beta b-4\alpha^2c=0$. It is exactly the same line, establishing tangency.
Back to article page
