- Identify the vertex as \((h, k) = (3, 2)\) and the directrix as \(y = 4\).
- Determine that the parabola opens downwards because the directrix is above the vertex, setting \(p = -2\).
- Use the standard form \((x - h)^2 = 4p(y - k)\) to write the equation \((x - 3)^2 = 4(-2)(y - 2)\).
- Conclude that the correct equation is \((x-3)^2 = -8(y-2)\), which corresponds to option a.
- Final answer: \(\boxed{\text{a}}\)
Explanation
- Analyzing the Given Information
The problem asks for the equation of a parabola given its vertex at \((3, 2)\) and its directrix at \(y = 4\). A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex \((h, k)\) of a parabola that opens vertically is located midway between the focus and the directrix. Given the vertex \((3, 2)\) and the horizontal directrix \(y = 4\), we can determine the orientation and the focal parameter \(p\).
- Determining the Equation Parameters
Since the directrix is a horizontal line \(y = 4\) and the vertex is at \((3, 2)\), the parabola must open vertically. Because the directrix \(y = 4\) lies above the vertex \(y = 2\), the parabola must open downwards. The standard form for a vertically opening parabola with vertex \((h, k)\) is \((x - h)^2 = 4p(y - k)\). Here, \(h = 3\) and \(k = 2\). The distance from the vertex to the directrix is given by \(|p|\). Since the directrix is \(y = k - p\), we have \(4 = 2 - p\), which means \(p = 2 - 4 = -2\). Since \(p\) is negative, this confirms the parabola opens downwards.
- Formulating and Selecting the Correct Equation
Now we substitute the values \(h = 3\), \(k = 2\), and \(p = -2\) into the standard form \((x - h)^2 = 4p(y - k)\):
\[(x - 3)^2 = 4(-2)(y - 2)\]
\[(x - 3)^2 = -8(y - 2)\]
Comparing this result with the given options:
A\((x-3)^2 = -8(y-2)\)
B\((x+3)^2 = 8(y+2)\)
C\((y-2)^2 = -4(x-3)\)
D\((y+2)^2 = 4(x+3)\)
The equation matches option a.
Examples
Parabolas are everywhere in our daily lives! For instance, satellite dishes are shaped as parabolas to focus incoming signals onto a single point. Similarly, the headlights of a car use a parabolic reflector to direct light into a strong, focused beam. By understanding the geometric properties of a parabola—its vertex, focus, and directrix—engineers can design these technologies to effectively capture and project signals or light. Understanding the equation \((x-3)^2 = -8(y-2)\) helps us mathematically model these physical shapes and their properties.