Conics Strike Again!

Geometry Level 5

The line \(x+y = 1 \) in \(xy\)-plane bisects two distinct chords of a standard parabola (which is symmetric about the \(x\)-axis and whose vertex is the origin having latus rectum length \(= 4a\)). If it is given that the intersection point of the two chords is \((a,2a)\), find the sum of all integral values of the possible lengths of latus rectum of the parabola.

Details and Assumptions

  • Latus rectum is the focal chord of the parabola which is perpendicular to its axis.
  • If your answers are 5, 6 and 7 then provide the answer as \(5+6+7 = 18\).
  • If you think no such parabola is possible give the answer as 0.
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