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).
2 Answers

“Assume a horizontal transverse axis and center (0, 0)”
Horizontal hyperbola centered at (0, 0):
x²/a² – y²/b² = 1
vertices (0±a, 0)
foci (0±c, 0), where c² = a² + b²
Use your data to solve for a and b
“vertex 4 units from center”
vertices (±4, 0)
a = 4
“focus 1 unit from mirror”
foci (±5, 0)
c = 5
c² = a² + b²
5² = 4² + b²
b = 3
The equation becomes
x²/4² – y²/3² = 1

Hyperbolic Mirror