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)

Explanation:

Step1: Determine the value of \(p\)

The distance between the focus \((3,9)\) and the vertex \((3,6)\) is \(p = 9 - 6=3\). Since the \(x\) - coordinates of the focus and vertex are the same, the parabola is vertical and its standard - form equation is \((x - h)^2=4p(y - k)\), where \((h,k)\) is the vertex. Here, \(h = 3\), \(k = 6\), and \(4p=12\). So the equation of the parabola is \((x - 3)^2=12(y - 6)\).

Step2: Rewrite the equations in standard form

For option A:
Starting with \(12y=x^{2}-6x + 81\), we can rewrite it as \(x^{2}-6x-12y + 81 = 0\). Completing the square for the \(x\) - terms: \(x^{2}-6x=(x - 3)^{2}-9\). Then \((x - 3)^{2}-9-12y + 81 = 0\), or \((x - 3)^{2}=12y-72\), which is not the correct form.
For option B:
Starting with \(24y=x^{2}-12x + 72\), we complete the square for the \(x\) - terms. \(x^{2}-12x=(x - 6)^{2}-36\). Then \((x - 6)^{2}-36+72 - 24y = 0\), or \((x - 6)^{2}=24y - 36\), which is not the correct form.
For option C:
Starting with \(24y=x^{2}-6x + 225\), we complete the square for the \(x\) - terms. \(x^{2}-6x=(x - 3)^{2}-9\). Then \((x - 3)^{2}-9+225-24y = 0\), or \((x - 3)^{2}=24y - 216\), which is not the correct form.
For option D:
The equation \((x - 3)^{2}=24(y - 9)\) has \(p = 6\) (since \(4p = 24\)), which is not correct as \(p = 3\) for our parabola.
For option E:
The equation \((x - 3)^{2}=12(y - 6)\) is in the correct form with \(p = 3\).
For option F:
The equation \((x - 9)^{2}=24(y - 3)\) has a vertex at \((9,3)\) which is not the correct vertex for our parabola.

Answer:

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