Cycling Conics

We need to use 2 facts:

For the general conic section equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ ,

• if its discriminant, $B^2-4AC = 0$ , then it is a parabola. And

• if the conic section is degenerate (an equation of pairs of intersecting lines), then the determinant of the matrix ${ \begin{bmatrix}{A} && {B} && {D} \\ {B} && {C} && {E} \\ {D} && {E} && {F}\end{bmatrix} }$ is zero. In other words, $\left(AC - \dfrac{B^2}4 \right) F + \frac14 (BED - CD^2 - AE^2) = 0.$

When $k =2$ , we get a parabola, $B^2 - 4AC = 0 \quad \Leftrightarrow\quad (Q+2)^2 - 4P^2 =0\quad \Leftrightarrow\quad Q + 2 = 2P$ When $k =4$ , $\left(AC - \dfrac{B^2}4 \right) F + \frac14 (BED - CD^2 - AE^2) = 0 \quad \Leftrightarrow\quad 3 P^3 - 8 P^2 - 11 P - 20 = 0$ Use rational root theorem to test for positive integer roots shows that $P = 4 \, (\text{only}) \Rightarrow Q = 6 \Rightarrow P+Q=\boxed{10}.$ For illustration, toggle the value of $k$ in Desmos to get a different conic sections.