UPSC NDA 2 2021 Mathematics Solved Paper

Concept:The endpoints of the latus rectum fix the focus and constrain the directrix direction.Explanation:Two fixed points are given as the endpoints of the latus rectum.The focus of a parabola is always the midpoint of the latus rectum.Therefore, the focus is uniquely determined and fixed.The length of the latus rectum is the distance between these two points.The distance from the focus to the directrix is half of the latus rectum length.Hence, this distance is also fixed.The directrix is always perpendicular to the axis of the parabola.Since the latus rectum is parallel to the directrix, the directrix must be parallel to the line joining the given endpoints.Thus, the slope of the directrix is fixed.However, the directrix can be placed on either side of the focus at that fixed perpendicular distance.This gives two possible parallel lines that can act as directrices.Each directrix, together with the fixed focus, defines a unique parabola passing through the given endpoints.Therefore, a maximum of two distinct parabolas can be drawn.Answer:Two (Option B).