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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.

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.

Grade:11

1 Answers

SAGAR SINGH - IIT DELHI
878 Points
13 years ago

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.

parabola

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