Question:
If the focus of a parabola is \((0,-3)\) and its directrix is
\[
y=3,
\]
then its equation is:
If the focus of a parabola is \((0,-3)\) and its directrix is
\[
y=3,
\]
then its equation is:
Show Hint
If the focus is \((0,-a)\) and the directrix is \(y=a\), then the vertex is at the origin and the parabola is \(x^2=-4ay\).
Updated On: Jun 18, 2026
- \(x^2=12y\)
- \(y^2=-12x\)
- \(y^2=12x\)
- \(x^2=-12y\)
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The Correct Option is D
Solution and Explanation
Step 1: Find the vertex of the parabola.
The focus is \[ (0,-3) \] and the directrix is \[ y=3. \] The vertex lies midway between the focus and the directrix.
Hence, \[ V=\left(0,\frac{-3+3}{2}\right)=(0,0). \]
Step 2: Determine the value of \(a\).
The distance from the vertex to the focus is \[ a=3. \] Since the focus lies below the vertex, the parabola opens downward.
Step 3: Use the standard equation.
For a parabola with vertex at the origin opening downward, \[ x^2=-4ay. \] Substituting \[ a=3, \] we get \[ x^2=-4(3)y. \] \[ x^2=-12y. \]
Step 4: Final conclusion.
Therefore, \[ \boxed{x^2=-12y} \]
The focus is \[ (0,-3) \] and the directrix is \[ y=3. \] The vertex lies midway between the focus and the directrix.
Hence, \[ V=\left(0,\frac{-3+3}{2}\right)=(0,0). \]
Step 2: Determine the value of \(a\).
The distance from the vertex to the focus is \[ a=3. \] Since the focus lies below the vertex, the parabola opens downward.
Step 3: Use the standard equation.
For a parabola with vertex at the origin opening downward, \[ x^2=-4ay. \] Substituting \[ a=3, \] we get \[ x^2=-4(3)y. \] \[ x^2=-12y. \]
Step 4: Final conclusion.
Therefore, \[ \boxed{x^2=-12y} \]
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