How many intersections can two confocal parabolas have? Can this be found out as a function of angle between the axes of the two parabolas? Also, what is the result in the case when the axes are perpendicular to each other?
I guessed the answer (two?) for the case in which the axes are inclined at $180°$, considering $y^2 = 4x$ and $y^2 = 8–4x$.
It'd be great if the solution could not only be mathematically rigourous but also based on properties of conics, and geometry.
Suppose that the common focus of the parabolae is $F$. Let $P$ be a point of intersection of the two parabolae. Then, by the geometry of parabolae, the distance from $F$ to $P$ is the same as the distance to either directrix (call them $d_1$ and $d_2$).
Recall that $P$ is equidistant to $d_1$ and $d_2$. Observe that a point is equidistant from two lines if it lies on the of the angle-bisector lines of the lines' intersection. Since the parabolae lie on one side of each directrix, there is one "quadrant" where the intersections can occur.
Therefore, the intersections lie on a single line, which can intersect to parabolae at most twice. Moreover, any intersection of this angle bisector must correspond to an intersection of the two parabolae.