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 $p$. Since the $x$-coordinates are the same, $p=9 - 6=3$. For a parabola with a vertical axis of symmetry (because $x$-coordinates of focus and vertex are equal), the 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 is $(x - 3)^2=12(y - 6)$.

Step2: Expand the standard - form equation

Expand $(x - 3)^2=12(y - 6)$. Using the formula $(a - b)^2=a^2-2ab + b^2$, we have $x^2-6x + 9=12y-72$. Rearranging gives $12y=x^2-6x + 81$.

Answer:

A. $12y=x^2-6x + 81$
E. $(x - 3)^2=12(y - 6)$