Assuming I have a general conic section $S\equiv ax^2+bxy+cy^2+dx+fy+e=0$ and line $L \equiv px+qy+r=0$, then what will be the general conic passing through their intersection points?
For instance, if I take $S+k \cdot L=0$, then this family of curves will not include all conics as the variable k does not affect the $x^2$, $y^2$ and $xy$ terms.
If I take $S+k \cdot L^2=0$, then this would contain only those curves which are tangent to the original conic section at the points of intersection. (because when we take $S+k \cdot L_1 L_2=0$ it is is the family of curves passing through points of intersection of $S$, $L_1$ and $L_2$. So when $L_1 \equiv L_2$, the curve will be tangent at the points of intersection of $S$ and $L_1$)