Two parabolas are orthogonal if the directrix of one is perpendicular to the other. Two orthogonal parabolas could intersect in 0, 1, 2, 3, or 4 points:
These five illustrations are GSP constructions based on the definition of a parabola as the locus of points equidistant from a line called the directrix and a point called the focus.
We could have also illustrated these examples with parabolas generated by equations:
(here we have omitted the focal points and the directrices)
For the case where orthogonal parabolas intersect in four points, show that the quadrilateral determined by those four points is cyclic. That is, there is a circle that lies on the four vertices of the quadrilateral.
1. Show this result via algebra
2. Show the result using a geometric argument.