1. A hyperbolic mirror can be used to take panoramic photos, if the camera is pointed toward the mirror with the lens at
one focus of the hyperbola. Write the equation of the hyperbola that can be used to model a mirror that has a vertex 5
inches from the center of the hyperbola and a focus 1 inch in front of the surface of the mirror. Assume the mirror has a
horizontal transverse axis and the hyperbola is centered at (0, 0).
2. Skip designs tracks for amusement park rides. For a new design, the track will be elliptical. If the ellipse is placed on a large coordinate grid with its center at (0, 0), the equation x^2/3600 + y^2/8100 = 1 models the path of the track. The units are given in yards. How long is the major axis of the track? Explain how you found the distance.
3. The equation of a parabola is 24y = (x - 2)^2 - 48. Identify the vertex, focus, and directrix of the parabola.
1. The equation of a hyperbola with a horizontal transverse axis and centered at (0, 0) is given by:
(x^2/a^2) - (y^2/b^2) = 1
Where a is the distance from the center to a vertex and c is the distance from the center to a focus. Given that the vertex is 5 inches from the center and the focus is 1 inch in front of the surface of the mirror, we have a = 5 and c = 6. Using the relationship c^2 = a^2 + b^2, we can solve for b:
b^2 = c^2 - a^2
b^2 = 6^2 - 5^2
b^2 = 36 - 25
b^2 = 11
Therefore, the equation of the hyperbola that can be used to model the mirror is:
(x^2/25) - (y^2/11) = 1
2. The equation of an ellipse with its center at (0, 0) is given by:
x^2/a^2 + y^2/b^2 = 1
Given the equation x^2/3600 + y^2/8100 = 1, we can see that a^2 = 3600 and b^2 = 8100. The major axis of the ellipse is twice the length of a, so the length of the major axis is:
2*sqrt(3600) = 2*60 = 120 yards
3. The general equation of a parabola in vertex form is:
4p(y-k) = (x-h)^2
Where the vertex of the parabola is at the point (h, k) and the focus is at the point (h, k+p). In this case, the vertex is at (2, 0) and the coefficient of y is 24, so 4p = 24 and p = 6. Therefore, the focus is at (2, 6) and the equation of the parabola is:
24y = (x-2)^2 - 48
The focus of the parabola is (2, 6) and the directrix is the line y = -6.