Hardcore Coordinate
Consider the set P={S=0 where S represents any conic with directrix , k is any real number} and an ellipse , if each member of P intersects the ellipse such that the common chord is of maximum length then find the set of values of k if P contains exactly one element, members of P are parabolas only .(answer upto two decimal only positive solution)
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The answer is 3.87.
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1 solution
Here's a hint: Translate the set-theoretic terminology into geometric figures (there will be only two such figures, as P contains exactly one such element defined by the given directrix); graph the figures in xy-coordinates, letting the slope of the directrix vary since we don't yet know the value of k; find the center of the ellipse; and then compute the common chord algebraically for different values of k. Through experiment, you may wonder what kind of property the parabola's directrix's slope must have in order to maximize the common chord's length.
Here's another hint: "Maximize" in this context usually means "optimize", which requires basic calculus.
Good luck!