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).
AAAaannndd the bot gets it wrong yet again!
The specifically stated that the center is at (0,0)
and the axis is horizontal, so the equation will be
x^2/a^2 - y^2/b^2 = 1
So fix it.
To write the equation of the hyperbola, we need to determine the distance from the vertex to the focus, which is also the distance from the vertex to the center of the hyperbola (since the hyperbola is centered at (0, 0)).
Using the distance formula, the distance from the vertex to the focus is:
sqrt((0-1)^2 + (0-4)^2) = sqrt(17)
Therefore, the equation of the hyperbola with vertex at (0, 4) and focus at (-1, 0) is:
((y-4)^2/16) - ((x+1)^2/17) = 1
Note that we used the negative value for the x-term because the focus is to the left of the vertex.