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 the focus lies on the axis of symmetry. The directrix is \(y = 3\), which is a horizontal line, so the axis of symmetry is the \(y\)-axis (vertical line \(x = 0\)).

Step2: Determine focus direction and distance

Since the directrix \(y = 3\) is above the vertex (origin), the parabola opens downward. The distance from the vertex to the directrix is \(|3 - 0|=3\). So the focus is 3 units below the vertex on the \(y\)-axis.

Step3: Find focus coordinates

The vertex is \((0,0)\), moving 3 units down (negative \(y\)-direction) gives the focus coordinates as \((0,0 - 3)=(0,-3)\).

Answer:

\((0,-3)\)