Question

If the focus of a parabola is \((0,-3)\) and its directrix is \[ y=3, \] then its equation is:

• A. \(x^2=12y\)
• B. \(y^2=-12x\)
• C. \(y^2=12x\)
• D. \(x^2=-12y\)

Hint:

If the focus is \((0,-a)\) and the directrix is \(y=a\), then the vertex is at the origin and the parabola is \(x^2=-4ay\).

Answer 1

The Correct Option is D

Step 1: Locate the vertex as the midpoint between focus and directrix.
Focus is (0, -3), directrix is y = 3. The vertex lies halfway: V = (0, (-3 + 3)/2) = (0, 0).

Step 2: Determine the focal distance a and orientation.
The distance from vertex to focus is a = 3. Since the focus is below the vertex, the parabola opens downward.

Step 3: Apply the standard equation for a downward-opening parabola.
For vertex at origin and opening downward: x² = -4ay. Substituting a = 3 gives x² = -12y.

Step 4: Final conclusion.
The equation of the parabola is x² = -12y.