Question

find the equation of the parabola with its focus at (0,-4) and its directrix y = 4.
a) y=-1/16x^2
b) y=-1/4x^2
c) y=-1/8x^2
d) y=1/16x^2

Explanation:

Step1: Determine the vertex

The vertex of a parabola is the mid - point between the focus $(0, - 4)$ and the directrix $y = 4$. The $x$ - coordinate of the vertex is $x = 0$. The $y$ - coordinate of the vertex is $\frac{-4 + 4}{2}=0$. So the vertex is at $(0,0)$.

Step2: Calculate the value of $p$

The distance $p$ from the vertex to the focus (or from the vertex to the directrix) is the absolute value of the difference between the $y$ - coordinate of the vertex and the $y$ - coordinate of the focus (or directrix). Here, $p=\vert0-(-4)\vert = 4$, and since the focus is below the vertex, the parabola opens downwards and $p=-4$.

Step3: Use the standard form of the parabola equation

The standard form of a parabola with vertex $(h,k)=(0,0)$ that opens up or down is $y - k=\frac{1}{4p}(x - h)^2$. Substituting $h = 0,k = 0,p=-4$ into the equation, we get $y=\frac{1}{4\times(-4)}x^{2}=-\frac{1}{16}x^{2}$.

Answer:

A. $y =-\frac{1}{16}x^{2}$