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Part 1
A telescope contains both a parabolic mirror and a hyperbolic mirror. They share focus Upper F 1, which is 43 feet above the vertex of the parabola. The hyperbola's second focus Upper F 2 is 9 ft above the parabola's vertex. The vertex of the hyperbolic mirror is 2 ft below Upper F 1. Find the equation of the hyperbola if the center is at the origin of a coordinate system and the foci are on the y-axis.
Given:
- The focus F1 is 43 feet above the vertex of the parabola, which means the y-coordinate of F1 is -43.
- The focus F2 is 9 feet above the vertex of the parabola, which means the y-coordinate of F2 is -9.
- The vertex of the hyperbolic mirror is 2 feet below F1, which means the y-coordinate of the vertex is -45.
Since the foci are on the y-axis and the center is at the origin, the equation of the hyperbola can be written in the form:
(y-k)^2/a^2 - (x-h)^2/b^2 = 1
Where:
- (h,k) is the center of the hyperbola (0,0 in this case)
- a is the distance from the center to the vertices of the hyperbola
- b is the distance from the center to the foci of the hyperbola
First, we need to find the value of a:
The distance between the vertices and the center is 2 ft (the vertex is 2 ft below F1), so a = 2.
Next, we need to find the value of b:
The distance between the foci and the center is 36 ft (43 - 9), so b = 36
Now we can plug in these values to get the equation of the hyperbola:
(y-0)^2/2^2 - (x-0)^2/36^2 = 1
y^2/4 - x^2/1296 = 1
Therefore, the equation of the hyperbola is y^2/4 - x^2/1296 = 1.