Question

Which equation represents a parabola that has a focus of (0, 0) and a directrix of y = 4? Responses x2=−2(y−2) ​ x squared equals negative 2 open parenthesis y minus 2 close parenthesis ​ x2=−8y x squared equals negative 8 y x2=−2y ​ x squared equals negative 2 y, , ​ x2=−8(y−2) x squared equals negative 8 open parenthesis y minus 2 close parenthesis

Answer 1

The standard form equation of a parabola with a vertical axis of symmetry is:

(x - h)^2 = 4p(y - k)

Where (h, k) is the vertex, p is the distance from the vertex to the focus (and from the vertex to the directrix), and the axis of symmetry is the line x = h.

In this case, the focus is at (0, 0), and the directrix is the line y = 4. Since the focus is at the origin, the vertex is also at the origin (h = 0, k = 0). The distance from the vertex to the directrix is 4, so the value of p is 4.

Therefore, the equation of the parabola is:

(x - 0)^2 = 4(4)(y - 0)

Simplifying this equation gives:

x^2 = 16y

So the correct equation that represents a parabola that has a focus of (0, 0) and a directrix of y = 4 is:

x^2 = 16y

Therefore, none of the given options represents the correct equation.