two parabolas have a common axis and concavities in opposite directions ; if any line parallel to the common axis meets the parabolas at p and p1 prove that locus of the mid point of pp1 is another parabola provided the latus recta of the given parabolas are unequal.

Dear student,

The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix).

In the following graph,

• The focus of the parabola is at (0, p).
• The directrix is the line y = -p.
• The focal distance is |p| (Distance from the origin to the focus, and from the origin to the directrix. We take absolute value because distance is positive.)
• The point (x, y) represents any point on the curve.
• The distance d from any point (x, y) to the focus (0, p) is the same as the distance from (x, y) to the directrix.