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).
Given that the hyperbola has a horizontal transverse axis and is centered at (0, 0), the equation of the hyperbola in standard form is:
(x^2 / a^2) - (y^2 / b^2) = 1
where a is the distance from the center to one vertex and c is the distance from the center to a focus.
Given that the vertex is 4 inches from the center and the focus is 1 inch in front of the mirror, c = 1 and a = √(c^2 + b^2).
The distance from the center to one vertex is always equal to a, so a = 4. Therefore, b = √(a^2 - c^2) = √(4^2 - 1^2) = √15.
Therefore, the equation of the hyperbola is:
(x^2 / 16) - (y^2 / 15) = 1