1169. Parabolae for disoriented people

    We were correcting parabola exercises. In some of them, the parabola axis was vertical, and in others, it was horizontal. This is always an added difficulty for disoriented students... Pepe Chapuzas, who is usually quite oriented, proposed the following challenge...    

    Dear Teacher:

    If we have 2 parabolae, one with horizontal axis and the other with vertical axis, that have 4 intersection points..., then there is a circumference passing through those 4 points.

    My well oriented student Quica Garrucha solved the exercise...

    Dear Teacher:

    I'm taking the axes of the two parabolae as coordinate axes. If  p  is the parameter of the parabola  P  and  q  is the parameter of the parabola  Q , then their equations are:

P : y² − 2px − m² = 0

Q : x² − 2qy − n² = 0

(P passes through (0, ±m) and Q passes through (±n, 0).)

    If we sum both equations we obtain

E : x² + y²− 2px − 2qy − m² − n² = 0

    E is a circumference with center  G (p, q)  and radius  r = √ (p²+q²+m²+n²) . The intersection points A, B, C, D, of the two parabolae satisfy the equations P and Q and therefore also E, that is, the circumference  E  passes through A, B, C and D.