# 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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