What do I get if my Directrix and Focus are the same?

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In my calculus class today we had a discussion about whether you'd get a vertical line or just a point when your directrix and focus become the same point. What would happen?

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You have everything zoomed out.At the nearest point the distance between focus and directrix is $ d(1+\epsilon)$ where d is distance to focus which when itself shrinks to zero entire scene goes to a point by uniform scaling. If length of feet of a person is 6 inches for his height 6 feet then those of an ant by same proportion would be 6 mm and 0.5 mm respectively.