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

Explanation:

Step1: Recall Parabola Properties

For a parabola, the directrix and focus are equidistant from the vertex, and the parabola opens towards the focus (away from the directrix). The standard form of a parabola that opens up or down (vertical axis) is \(x^{2}=4py\), where the directrix is \(y = -p\) and the focus is \((0,p)\). If it opens left or right (horizontal axis), the standard form is \(y^{2}=4px\), directrix \(x=-p\), focus \((p,0)\).

Step2: Analyze Directrix \(y = 3\)

The directrix is a horizontal line (\(y = 3\)), so the parabola opens vertically (up or down). For vertical parabola \(x^{2}=4py\), directrix is \(y=-p\). Here, directrix \(y = 3=-p\), so \(p=- 3\).

Step3: Find Focus and Equation

If \(p=-3\), the focus is \((0,p)=(0, - 3)\). The equation is \(x^{2}=4py=4\times(-3)y=-12y\).

Answer:

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