Question

question 3 (2 points)
(02.08)
derive the equation of the parabola with a focus at (-5, -5) and a directrix of y = 7.
o a
$f(x)=-\frac{1}{24}(x - 1)^2-5$
o b
$f(x)=\frac{1}{24}(x - 1)^2-5$
o c
$f(x)=-\frac{1}{24}(x + 5)^2+1$
o d
$f(x)=\frac{1}{24}(x + 5)^2+1$

Explanation:

Step1: Find the vertex

The vertex of a parabola is the mid - point between the focus $(-5,-5)$ and the point on the directrix directly above/below the focus. The $x$ - coordinate of the vertex is the same as the $x$ - coordinate of the focus, $x=-5$. The $y$ - coordinate of the vertex is $\frac{-5 + 7}{2}=1$. So the vertex $(h,k)=(-5,1)$.

Step2: Determine the value of $a$

The distance $p$ between the vertex and the focus (or the vertex and the directrix) is $|1-(-5)| = 6$. For a parabola, the equation is of the form $y=a(x - h)^2+k$, and $a=\frac{1}{4p}$. Since the focus is below the directrix, the parabola opens downwards and $a=-\frac{1}{24}$.

Step3: Write the equation of the parabola

Substitute $h=-5$, $k = 1$ and $a=-\frac{1}{24}$ into the equation $y=a(x - h)^2+k$. We get $y=-\frac{1}{24}(x + 5)^2+1$, or $f(x)=-\frac{1}{24}(x + 5)^2+1$.

Answer:

C. $f(x)=-\frac{1}{24}(x + 5)^2+1$