Question

A camera is placed in front of a hyperbolic mirror. The equation of the hyperbola that models the mirror is y^2/16-x^2/9=1 where x and y are in inches the camera is pointed toward the vertex of the hyperbolic mirror and is positioned such that the lens is at the nearest focus to that vertex. The lens is_____ inch(es) from the mirror.

Answer 1

Answer:

The lens is 6 inches from the mirror

Step-by-step explanation:

We need to find the distance of the lens from the mirror, if  the camera is pointed toward the vertex of the hyperbolic mirror and is positioned such that the lens is at the nearest focus to that vertex.

Given:

$\frac{y^2}{16} - \frac{x^2}{9}=1$

where a² = 16 and b² = 9

Then c² = a² + b²

c² = 16+9

c² = 25

c= √25 = 5

The hyperbola is centered at (0,0)

The vertex is at (0,a) = (0,4)

The foci is at (0,c) = (0,5)

The distance from vertex to focus is

d= c-a = 5-4 = 1 inches

Total distance from lens to mirror is sum of distance d and foci c:

5+1= 6 inches