Question

note: Enter your answer and show all the steps that you use to solve this problem in the space provided.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 4 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). show all work

Answer 1

The standard form equation of a hyperbola centered at (0,0) with horizontal transverse axis is:

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

Where:
a = distance from center to vertex
c = distance from center to focus

Given:
a = 4 inches
c = 1 inch

Since c^2 = a^2 + b^2 for a hyperbola:
1^2 = 4^2 + b^2
1 = 16 + b^2
b^2 = -15

Therefore, the equation of the hyperbola is:
(x^2/16) - (y^2/-15) = 1

Which simplifies to:
x^2/16 + y^2/15 = 1