Question

A telescope contains both a parabolic mirror and a hyperbolic mirror. They share focus (F_1), which is 46 feet above the vertex of the parabola. The hyperbola's second focus (F_2) is 8 ft above the parabola's vertex. The vertex of the hyperbolic mirror is 2 ft below (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.

A. (y^2 - x^2 = 160)
B. (x^2 - y^2 = 160)
C. (y^2 + x^2 = 160)
D. (x^2 + y^2 = 160)

Answer 1

The equation of the hyperbola is (x^2 - y^2 = 160).

Step-by-step explanation:

The equation of the hyperbola can be found using the standard form for a hyperbola with its center at the origin and foci on the y-axis: y^2/a^2 - x^2/b^2 = 1. To find the values of a and b, we can use the given information. Let's denote the distance between F1 and the vertex of the hyperbola as c. Since F1 is 46 ft above the vertex of the parabola, c = 46. The distance between F2 and the vertex of the hyperbola is 2 ft + c = 48 ft. We know that the distance between the foci, 2c, is equal to 8 ft, so 2c = 8. From this, we can solve for c, and then find the values of a and b. The correct answer is option B: (x^2 - y^2 = 160).