Question

Choose ,begin emphasis,all,end emphasis, of the equations that represent a parabola with the focus the point 3 comma 9 and the vertex the point 3 comma 6.

Answer options with 6 options
A.
12 y equals x squared minus 6 x plus 81

B.
24 y equals x squared minus 12 x plus 72

C.
24 y equals x squared minus 6 x plus 225

D.
left-parenthesis x minus 3 right-parenthesis squared equals 24 left-parenthesis y minus 9 right-parenthesis

E.
left-parenthesis x minus 3 right-parenthesis squared equals 12 left-parenthesis y minus 6 right-parenthesis

F.
left-parenthesis x minus 9 right-parenthesis squared equals 24 left-parenthesis y minus 3 right-parenthesis

Answer 1

To find the equation of a parabola with a given focus and vertex, we should start by understanding the basic properties of parabolas.

**Given:**
- Focus: (3, 9)
- Vertex: (3, 6)

Since the focus is above the vertex (both points have the same x-coordinate), this parabola opens upwards. The standard form of the equation for a parabola that opens upwards is:

\[
(x - h)^2 = 4p(y - k)
\]

where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.

**Calculating \(p\):**
The vertex is at (3, 6) and the focus is at (3, 9). The distance \(p\) is:

\[
p = 9 - 6 = 3
\]

In this case, since the parabola opens upward, we have:

\[
4p = 4 \times 3 = 12
\]

**Substituting into the equation:**

Using \((h, k) = (3, 6)\) and \(4p = 12\), the equation becomes:

\[
(x - 3)^2 = 12(y - 6)
\]

Now, let’s check the available options for a match:

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

The equation we derived was \((x - 3)^2 = 12(y - 6)\).

Now we can identify the correct options:

**The correct option is:**
**E.** \((x - 3)^2 = 12(y - 6)\)

This matches our derived equation.