Hardcore Coordinate

Geometry Level 5

Consider the set P={S=0 where S represents any conic with directrix \(x+ky+16=0\) , k is any real number} and an ellipse \(E = 9x^{2}+16y^{2}=144\), 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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