Intersection of a conic and a line

Question

Suppose $C=\{x_0^2+x_1^2+x_2^2\}\subset \mathbb{P}^2_{x_0,x_1,x_2}$, and $L=\{a_0x_0+a_1x_1+a_2x_2\}\subset \mathbb{P}^2_{x_0,x_1,x_2}$, I was asked to find the set of coefficients $(a_0,a_1,a_2)$ such that $C$ and $L$ do not intersect at two distinct points. Isn't $C$ only contain the origin in this case? How can I find such coefficients?

Answer 1

The main idea here is that a conic and a line always meet at two points when counted with multiplicity. In other words, you are looking for triples of coefficients such that the line meets $C$ with multiplicity two, i.e. is tangent to $C$. This is basically a multivariable calculus exercise (hint: you should get a condition of the form $F = 0$, where $F$ is a homogeneous polynomial in $a_0,a_1,$ and $a_2$).