Question

a lighthouse is located at (1, 2) in a coordinate system measured in miles. a sailboat starts at (-7, 8) and sails in a positive x - direction along a path that can be modeled by a quadratic function with a vertex at (2, -6). which system of equations can be used to determine whether the boat comes within 5 miles of the lighthouse?
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$$\begin{cases}(x - 1)^2 + (y - 2)^2 = 5 \\\\ y = \\frac{14}{81}(x - 2)^2 - 6\\end{cases}$$

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$$\begin{cases}(x - 1)^2 + (y - 2)^2 = 25 \\\\ y = \\frac{14}{81}(x - 2)^2 - 6\\end{cases}$$

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$$\begin{cases}(x - 1)^2 + (y - 2)^2 = 5 \\\\ y = -\\frac{14}{81}(x + 7)^2 + 8\\end{cases}$$

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$$\begin{cases}(x - 1)^2 + (y - 2)^2 = 25 \\\\ \\dots \\frac{14}{\\dots}(\\dots + 7)^2 \\dots \\end{cases}$$

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Explanation:

Step1: Determine the circle equation

The lighthouse is at \((1, 2)\) and we want to find when the boat is within 5 miles. The equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2+(y - k)^2=r^2\). Here, \(h = 1\), \(k = 2\), and \(r = 5\), so \(r^2=25\). Thus, the circle equation is \((x - 1)^2+(y - 2)^2 = 25\).

Step2: Determine the quadratic equation

The boat starts at \((-7, 8)\) and has a vertex at \((2,-6)\). The vertex form of a quadratic is \(y=a(x - h)^2+k\), where \((h, k)\) is the vertex. So \(h = 2\), \(k=-6\), so \(y=a(x - 2)^2-6\). We use the point \((-7, 8)\) to find \(a\):

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So the quadratic equation is \(y=\frac{14}{81}(x - 2)^2-6\).

Answer:

The second option: \(

$$\begin{cases}(x - 1)^2+(y - 2)^2 = 25\\y=\frac{14}{81}(x - 2)^2-6\end{cases}$$

\) (the option with the circle equation \((x - 1)^2+(y - 2)^2 = 25\) and the quadratic \(y=\frac{14}{81}(x - 2)^2-6\))