Question

if the directrix of a parabola is the horizontal line y = 3, what is true of the parabola? the focus is at (0, 3), and the equation for the parabola is y² = 12x. the focus is at (0, -3), and the equation for the parabola is x² = -12y. the focus is at (3, 0), and the equation for the parabola is x² = 12y. the focus is at (-3, 0), and the equation for the parabola is y² = -12x.

Explanation:

Step1: Recall parabola property

For a parabola, the directrix \(y = k - p\) and focus \((0,k + p)\) (for parabola \(x^{2}=4py\)) or directrix \(x = h - p\) and focus \((h + p,0)\) (for parabola \(y^{2}=4px\)). Given directrix \(y = 3\), the parabola is of the form \(x^{2}=- 4py\) (opens down - ward) and the focus is on the \(y\) - axis. The distance from the vertex to the directrix is \(p\). The vertex of the parabola is halfway between the focus and the directrix. For a parabola with directrix \(y = 3\), if the vertex is at the origin \((0,0)\), then the focus is at \((0,- 3)\) since the distance from the vertex to the directrix and from the vertex to the focus is the same.

Step2: Determine the equation

The standard form of a parabola with vertex \((0,0)\) and focus \((0,-p)\) is \(x^{2}=-4py\). Since the distance from the vertex \((0,0)\) to the directrix \(y = 3\) is \(p = 3\), the equation is \(x^{2}=-12y\).

Answer:

The focus is at \((0,-3)\), and the equation for the parabola is \(x^{2}=-12y\).