Analyze this conic section to answer the questions below. \( x^{2}=16(y-6) \) Parabola Ellipse Hyperbola Where is the center of this conic section? Select the correct choice below and fill in any answer boxes in your choice. A. Type an ordered pair.) What are the values of a and b for this conic section? Select the correct choice below and fill in any answer boxes in your choice. A. a = and b does not have a value B. a does not have a value anlibo =

We start with the given equation:   x² = 16(y – 6) Step 1. Identify the Conic Section The equation is of the form   (x – h)² = 4p (y – k) which describes a parabola that opens either upward (if p > 0) or downward (if p < 0). Here h = 0 and k = 6, and comparing:   4p = 16 → p = 4. So the parabola opens upward. Thus, the correct choice is: Parabola. Step 2. Center (or Vertex) of the Conic Section In a parabola, there is no “center” like there is for ellipses or hyperbolas; it has a vertex which serves as the “turning point” of the graph. In the equation above, the vertex is located at (h, k), which is (0, 6). So the correct response about the center is to note that a parabola does not have a center. (If your answer box expects an ordered pair for a “center”, you should instead indicate that the parabola has a vertex at (0, 6), not a center.) Step 3. Values of a and b for This Conic Section For ellipses and hyperbolas, we typically define both a and b. However, for a parabola, only one parameter (related to the distance from the vertex to the focus, often denoted as p) is defined; there is no second parameter b. The given answer choices are:  A. a = ___ and b does not have a value  B. a does not have a value and b = ___ Because we are working with a parabola, we choose option A. Here, the coefficient 16 corresponds to 4p so that p = 4. Since some texts may denote “a” as the distance parameter for a parabola (though the identifying letter is usually p), we interpret the answer as:   a = 4 and b does not have a value. Final Answers: 1. The conic is a Parabola. 2. There is no center; the vertex is (0, 6). 3. Option A is correct: a = 4 and b does not have a value.