Question

Choose all of the equations that represent a parabola with the focus (3, 9) and the vertex (3, 6).

A. 12y = x ^ 2 - 6x + 81

B. 24y = x ^ 2 - 12x + 72

C. 24y = x ^ 2 - 6x + 225

D. (x - 3) ^ 2 = 24(y - 9)

E. (x - 3) ^ 2 = 12(y - 6)

F. (x - 9) ^ 2 = 24(y - 3)

Answer 1

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.

1. **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)**