Analyze this conic section to answer the questions below. \( x^{2}=16(y-6) \) B. This conic section has a vertex at \( (2,4) \). C. This conic section has vertices at \( (21,0) \) and \( (2,0) \). D. This conic section has vertices at \( (4,0) \) and \( (-4,0) \). What is the value of c for this conic section? Select the correct choice below and fill in any answer boxes in your choice. (Simplify your answer. Type an exact answer, using radicals as needed.) Determine the focus or foci for the conic section. A. The foci for this conic section are at \( (-\sqrt{21}, 0) \) and \( (\sqrt{21}, 0) \). B. The focus for this conic section is at \( (0, \sqrt{21}) \). O. The focus for this conic section is at \( (0,10) \). O D. The foci for this conic section are at \( (4,0) \) and \( (21,0) \).

To break down the given conic section \( x^{2} = 16(y - 6) \), we can identify that this is a vertical parabola. The vertex form reveals that the vertex is at \( (0, 6) \) since it represents the form \( x^2 = 4p(y - k) \) where \( (h, k) \) is the vertex. Now, for the second part of your question: - The value of \( c \) is calculated from the equation, where \( c = \sqrt{p^{2} + k^{2}} \). Here, \( p \) (the distance from the vertex to the focus) equals 4, hence \( c = 4 \). And regarding the focus: - The focus is located above the vertex at \( (0, 10) \) by moving 4 units up from the vertex. Thus, the correct responses would be: C. The focus for this conic section is at \( (0,10) \). Now for some added fun knowledge: Did you know that the word "parabola" comes from the Greek word "parabolē," which means "a comparison"? It perfectly encapsulates how parabolas translate geometric concepts into algebraic expressions, creating a beautiful connection between different math realms! Also, parabolas are not just limited to mathematical studies; they are found everywhere in the real world! The paths of tossed objects (like basketballs or cannonballs) trace out parabolic shapes, which is why understanding their equations can help in fields like physics and engineering!