To determine which equations represent a parabola with the given focus (3, 9) and vertex (3, 6), we can start by identifying some key properties of parabolas.
The vertex form of a parabola that opens upwards or downwards is given by:
\[ (x - h)^2 = 4p(y - k) \]
where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.
- Identify the parameters:
- The vertex \((h, k) = (3, 6)\)
- The focus is at \((3, 9)\)
- The distance \(p\) from the vertex to the focus is \(9 - 6 = 3\).
Since the focus is above the vertex, the parabola opens upwards. In this case, \(4p = 4 \times 3 = 12\).
Therefore, the equation of the parabola is:
\[ (x - 3)^2 = 12(y - 6) \]
Now, we can analyze each option provided:
A. \(12y = x^2 - 6x + 81\)
Rearranging gives: \[ x^2 - 6x - 12y + 81 = 0 \] This is not in the standard form of a parabola that opens up or down.
B. \(24y = x^2 - 12x + 72\)
Rearranging gives: \[ x^2 - 12x - 24y + 72 = 0 \] Again, not in standard form.
C. \(24y = x^2 - 6x + 225\)
Rearranging gives: \[ x^2 - 6x - 24y + 225 = 0 \] Still not in standard form.
D. \((x - 3)^2 = 24(y - 9)\)
This has \(4p = 24\), which means \(p = 6\). The vertex is (3, 9) and focuses above the vertex (3, 15). This does not match the vertex.
E. \((x - 3)^2 = 12(y - 6)\)
This is exactly the equation we derived based on the vertex and focus! This is a valid equation.
F. \((x - 9)^2 = 24(y - 3)\)
Here, \(k = 3\) which indicates the vertex is (9, 3), not (3, 6). This does not match our vertex.
Conclusion: The only equation that represents a parabola with the focus (3, 9) and the vertex (3, 6) is:
E. (x - 3) ^ 2 = 12(y - 6)