A telescope contains both a parabolic mirror and a hyperbolic mirror. They share focus Upper F 1, which is 44 feet above the vertex of the parabola. The hyperbola's second focus Upper F 2 is 6 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.

Question

A telescope contains both a parabolic mirror and a hyperbolic mirror. They share focus Upper F 1​, which is 44 feet above the vertex of the parabola. The​ hyperbola's second focus Upper F 2 is 6 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.
Question content area bottom left
Part 1
Complete the equation.
  
enter your response hereminus
  
enter your response hereequals1

GPT 3.5 · Answer

The equation of the hyperbola is:

y^2 / a^2 - x^2 / b^2 = 1

where:
- center is at the origin, so h = 0 and k = 0
- foci are on the y-axis, so a = 6
- c = distance from center to focus, so c = 6
- distance from vertex to focus is 2 ft, so b = 2

Now, we can find the value of b^2 using the relationship between a, b, and c in a hyperbola:

c^2 = a^2 + b^2
6^2 = 6^2 + b^2
36 = 36 + b^2
b^2 = 0

Therefore, the equation of the hyperbola is y^2 / 36 - x^2 / 0 = 1, which simplifies to y^2 / 36 = 1, or y^2 = 36.