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Prove some properties of parabolas

  1. A chord with length 8 |c| is perpendicular to the axis of the parabola y^2 = 4cx. If P and Q are the points where the chord and the parabola meet, prove that the vector from O to P is perpendicular to the vector from O to Q.
  2. Show that the length of the chord of a parabola drawn through the focus and parallel to the directrix (the latus rectum) is twice the distance from the focus to the directrix. Next, show that the tangents to the parabola at both ends of this chord intersect the axis of the parabola on the directrix.

Incomplete.

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