Answer: [tex]\frac{x^2}{1600} - \frac{y^2}{900} = 1[/tex]
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Explanation:
The two focal points are always found on the major axis of the hyperbola. The minor axis runs perpendicular to the major axis.
The two focal points are (-50,0) and (50,0). The midpoint of these focal points is (0,0) and this represents the center of the hyperbola. So (h,k) = (0,0)
These focal points lie along the x axis, meaning each curve opens in a horizontal direction. The "parabola" sub pieces open left/right or east/west. Based on this configuration, we'll use this template
[tex]\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1[/tex]
The more general template is
[tex]\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1[/tex]
but we can ignore the h,k because they're both 0.
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So the key is to find the values of 'a' and b.
We're told that the difference in distances is 80 miles. The second hint says that 2a represents this difference. So 2a = 80 leads to a = 40.
To find b, we apply the pythagorean theorem. The 'c' value represents the distance from the center to either focus. In this case, it would be c = 50.
[tex]a^2+b^2 = c^2\\\\b^2 = c^2-a^2\\\\b = \sqrt{c^2-a^2}\\\\b = \sqrt{50^2-40^2}\\\\b = \sqrt{2500-1600}\\\\b = \sqrt{900}\\\\b = 30\\\\[/tex]
We can therefore update the template we were working with earlier
[tex]\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\\\\\frac{x^2}{40^2} - \frac{y^2}{30^2} = 1\\\\\frac{x^2}{1600} - \frac{y^2}{900} = 1\\\\\left(\frac{x}{40}\right)^2 - \left(\frac{y}{30}\right)^2 = 1\\\\[/tex]
The graph is shown below.
