Question

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).

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Answer 1

To find the equation of the hyperbola, we first need to find the coordinates of the vertex and focus.

Given:
Vertex = (4, 0)
Focus = (-1, 0)

The distance from the center to the vertex (c) is the distance from the center to the focus:
c = 4 - (-1) = 5

The distance from the center to a focus is related to the distance from the center to a vertex by the hyperbola formula:
c^2 = a^2 + b^2
5^2 = 4^2 + b^2
25 = 16 + b^2
b^2 = 9
b = 3

Since the hyperbola is centered at (0, 0) and has a horizontal transverse axis, the equation of the hyperbola is:
x^2/a^2 - y^2/b^2 = 1

Plugging in the values of a and b:
x^2/16 - y^2/9 = 1

Therefore, the equation of the hyperbola is:
x^2/16 - y^2/9 = 1