Finding a relation between l, m, n given $a_1l+b_1m+c_1n=d_1$ and $a_2l^2+b_2m^2+c_2n^2=d_2$

In quite a few analytic geometry questions, we need to find the relation between l,m and n given the set of equations :

$$a_1l+b_1m+c_1n=d_1$$ $$a_2l^2+b_2m^2+c_2n^2=d_2$$

Is there a general approach we can use to find such a relation?

My approach was to eliminate n from the first equation and to get an equation between l and m, however I ended up with a 2 degree polynomial of l and m and I cannot factorize it down to linear terms. Is there a better way to do it?

• In the $l,m,n$ space the two given equations are those of a plane and a quadric respectively. Their intersection is a conic (it can be an ellipse, a hyperbola, or a parabola). This conic will lie in the plane specified by the first equation. You can attach a reference frame $Oxyz$ whose $xy$ plane corresponds to the plane given by the first equation. Then re-write the second equation using $x,y,z$. Now setting $z=0$ gives a relation between $x$ and $y$. Finally, from this relation, you can retrieve $(l,m,n)$ from $(x,y,0)$. Commented Sep 5 at 8:15
• For it to factor into linear terms, the intersection should be a pair of lines - which is not always the case, so you shouldn't expect the factorisation to work in general. Commented Sep 7 at 4:02