Question

select the correct answer.
what is the equation of a parabola with a focus at (-6,0) and a directrix of y = 8?
○ $y = -\frac{1}{16}(x + 6)^2 + 4$
○ $y = \frac{1}{32}(x + 6)^2$
○ $y = -\frac{1}{32}(x + 6)^2$
○ $y = \frac{1}{16}(x + 6)^2 + 4$

Explanation:

Step1: Find the vertex

The vertex \((h,k)\) of the parabola is the midpoint between the focus \((-6,0)\) and the directrix \(y = 8\). The \(x\)-coordinate of the vertex is the same as the focus's \(x\)-coordinate, so \(h=-6\). The \(y\)-coordinate \(k\) is the average of the \(y\)-coordinate of the focus and the directrix: \(k=\frac{0 + 8}{2}=4\). So the vertex is \((-6,4)\).

Step2: Determine the direction and \(p\)

Since the directrix is \(y = 8\) (horizontal line) and the focus is below the directrix, the parabola opens downward. The distance \(p\) between the vertex and the focus (or directrix) is \(|4 - 0|=4\) (but since it opens downward, \(p=-4\)). Wait, no: the formula for a parabola that opens up or down is \((x - h)^2=4p(y - k)\). The distance between focus \((-6,0)\) and vertex \((-6,4)\) is \(|4 - 0| = 4\), so \(p=0 - 4=-4\) (because focus is below vertex, \(p\) is negative for downward opening).

Step3: Write the standard form and convert to vertex form

The standard form is \((x - (-6))^2=4p(y - 4)\), so \((x + 6)^2=4(-4)(y - 4)\), which simplifies to \((x + 6)^2=-16(y - 4)\). Now, solve for \(y\):

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Wait, looking at the options, one of the options is \(y = -\frac{1}{16}(x + 6)^2+4\) (first option? Wait the options: first option is \(y = -\frac{1}{16}(x + 6)^2+4\)? Wait the user's options: let's re - check. The first option is \(y=-\frac{1}{16}(x + 6)^2 + 4\)? Wait the user's image: the first option is \(y = -\frac{1}{16}(x + 6)^2+4\)? Wait the options as per the image (from the user's input):

Option 1: \(y=-\frac{1}{16}(x + 6)^2 + 4\)

Option 2: \(y=\frac{1}{32}(x + 6)^2\)

Option 3: \(y =-\frac{1}{32}(x + 6)^2\)

Option 4: \(y=\frac{1}{16}(x + 6)^2+4\)

Wait, maybe I made a mistake in calculating \(p\). Let's recalculate \(p\). The distance between the focus \((-6,0)\) and directrix \(y = 8\) is \(|8 - 0| = 8\), so the distance from vertex to focus (or directrix) is \(\frac{8}{2}=4\). So \(p\) is the distance from vertex to focus: vertex is \((-6,4)\), focus is \((-6,0)\), so \(p=0 - 4=-4\). Then \(4p=4\times(-4)=-16\). So the equation \((x + 6)^2=4p(y - 4)\) becomes \((x + 6)^2=-16(y - 4)\), then \(y-4=-\frac{1}{16}(x + 6)^2\), so \(y =-\frac{1}{16}(x + 6)^2+4\), which matches the first option (if the first option is \(y = -\frac{1}{16}(x + 6)^2+4\)).

Answer:

\(y = -\frac{1}{16}(x + 6)^2+4\) (the first option among the given options)