Question

a parabola has a vertex at the origin. the equation of the directrix of the parabola is y = 3. what are the coordinates of its focus? (0,3) (3,0) (0,−3) (−3,0)

Explanation:

Step1: Recall parabola focus-directrix property

For a parabola with vertex at the origin \((0,0)\), the focus and directrix are equidistant from the vertex, and lie on the axis of symmetry. The directrix is \(y = 3\) (a horizontal line? No, \(y=3\) is a horizontal line? Wait, \(y = k\) is horizontal, \(x = k\) is vertical. The directrix \(y=3\) is a horizontal line, so the parabola opens up or down. The axis of symmetry is the \(y\)-axis (since vertex is at origin and directrix is horizontal, axis is \(x = 0\) (the \(y\)-axis)).

Step2: Determine direction and focus position

The directrix \(y = 3\) is above the vertex (origin). For a parabola, the focus is on the opposite side of the vertex from the directrix. So if directrix is \(y = 3\) (above origin), the parabola opens downward, and the focus is below the vertex (origin) on the \(y\)-axis. The distance from vertex to directrix is \(|3 - 0|=3\), so the focus is 3 units below the vertex. The vertex is at \((0,0)\), so moving 3 units down (along \(y\)-axis, \(x = 0\), \(y=0 - 3=-3\)) gives the focus coordinates \((0, - 3)\).

Answer:

C. \((0,-3)\) (assuming the options are labeled as A: \((0,3)\), B: \((3,0)\), C: \((0,-3)\), D: \((-3,0)\))