Respuesta :
Answer:
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
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:
- [tex]The \ equation \ of \ the \ hyperbola \ that \ models \ the \ mirror \ is \ \underline{\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1}[/tex]
Details of the method used to find the equation
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;
- [tex]\mathbf{\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2}} = 1[/tex]
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;
- [tex]\dfrac{(x - 0)^2}{4^2} - \dfrac{(y - 0)^2}{3^2} = \underline{ \dfrac{x^2}{16} - \dfrac{y^2}{9} = 1}[/tex]
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