Ahyperbolic 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).

The distance of the focus from the center is the sum of the distance from

the focus to the surface and the vertex distance.

Correct response:

Given:

Distance of the vertex from the center = 4 inches

Distance of the focus from the mirror surface = 1 inches

Coordinates of the center of the mirror = (0, 0)

Required:

To write the equation of the hyperbola that can be used to model the mirror

Solution:

The equation of an hyperbola having an horizontal transverse axis is presented as follows;

Where;

(h, k) = The coordinate of the center

a = Center to vertex distance

b² = c² - a²

Where;

c = The from the center to the vertex

Therefore;

a = 4

(h, k) = (0. 0)

c = 4 + 1 = 5

b² = 5² - 4² = 9

b = √9 = 3

The equation of the hyperbola is therefore;

Learn more about the equation of a hyperbola here:

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The equation of the hyperbola is x²/16 - y²/9 = 1

Step-by-step explanation:

* Lets study the equation of the hyperbola

- The standard form of the equation of a hyperbola with  

  center (0 , 0) and transverse axis parallel to the x-axis is

  x²/a² - y²/b² = 1

- The length of the transverse axis is 2a

- The coordinates of the vertices are (±a , 0)

- The length of the conjugate axis is 2b

- The coordinates of the co-vertices are (0 , ±b)

- The coordinates of the foci are (± c , 0),  

- The distance between the foci is 2c where c² = a² + b²

- The distance between the vertex and the focus in-front of it is c - a

* Now lets solve the problem

- The distance from a vertex to the center of the mirror

∵ The vertex of the mirror is (a , 0)

∵ The distance between a vertex and the center of the mirror

  is 4 inches

∴ a = 4

∵ The distance between the vertex and a focus in front of  the surface

  of the mirror is 1

∵ The distance between the vertex and the focus in-front of it is c - a

∴ c - a = 1

∴ c - 4 = 1 ⇒ add 4 to the both sides

∴ c = 5

- The mirror has a horizontal transverse axis and the hyperbola is

 centered at (0, 0)

∴ The equation of the hyperbola is x²/a² - y²/b² = 1

- Lets find b from a and c

∵ c² = a² + b²

∵ c = 5 and a = 4

∴ (5)² = (4)² + b²

∴ 25 = 16 + b² ⇒ subtract 16 from both sides

∴ 9 = b² ⇒ take √ for both sides

∴ b = ±3

- Lets write the equation

∴ x²/(4)² - y²/(3)² = 1

∴ x²/16 - y²/9 = 1

* The equation of the hyperbola is x²/16 - y²/9 = 1