Finding all points of intersection between a line and a nonsingular conic $C$

Question

This is the same conic, $C$ as the one in this question with symmetric matrix represenatation \begin{pmatrix} 3 & 7/2 & 2\\ 7/2 & 2 & 5/2\\ 2 & 5/2 & 3\end{pmatrix}

I am trying to find all points of intersection between the line $x = 0$ and $C$.

To do this do I use Joachimsthal's equation where $A$ is the symmetric matrix and the line $l$ is represented in parametric with a point $P+\lambda Q$ that lies on $C$ when: $$P^{t}AP +2\lambda P^{t}AQ + \lambda^{2}Q^{t}AQ =0 $$.

What do I use for points $P$ and $Q$?

Answer 1

To describe the line $x=0$, any two distinct points on that line may be used as $P$ and $Q$. So pick two vectors of the form $(0,*,*)^t$, as simple as possible. The simplest I can think of would be $P=(0,0,1)^t$ and $Q=(0,1,0)^t$. If you homogenize using $z=1$, as is the convention, then the former would denote the origin, the latter the point at infinity in $y$ direction.

Using these, you can read the terms of your equation from the matrix fairly easily. $P^tAP=3$ is the entry in the bottom right corner. $Q^tAQ=2$ the central element. And $2P^tAQ=5$ the sum $A_{23}+A_{32}$. So your equation reads

$$3+5\lambda+2\lambda^2=0$$

From this you can read $\lambda\in\{-1,-\frac32\}$. Then $P+\lambda Q=(0,\lambda,1)^t$ so these values are the $y$ coordinates of your points of intersection.

Answer 2

I had never heard of Joachimsthal's equation before, and I'm not sure why you're writing the equation of the conic in matrix form. Here is a different approach.

A point of intersection $(x,y)$ of the line given by $x=0$ and the conic given by $$ 3x^{2}+ 2y^{2}+ 7xy + 4x + 5y + 3 = 0 $$ must satisfy both of these equations. The equation for the line implies $x=0$, and substituting this into the equation for the conic yields $$ 2y^2 + 5y + 3 = 0 \, . $$ The two solutions $y_1, y_2$ to the above quadratic equation will give you your points of intersection $(0, y_1), (0, y_2)$.