Answer:
The vertices are (3 , -5) , (-5 , -5)
The foci are (4 , -5) , (-6 , -5)
Step-by-step explanation:
* Lets study the equation of the hyperbola
- The standard form of the equation of a hyperbola with
center (h , k) and transverse axis parallel to the x-axis is
(x - h)²/a² - (y - k)²/b² = 1
- The length of the transverse axis is 2 a
- The coordinates of the vertices are (h ± a , k)
- The coordinates of the foci are (h ± c , k), where c² = a² + b²
- The distance between the foci is 2c
* Now lets solve the problem
- The equation of the hyperbola is (x + 1)²/16 - (y + 5)²/9 = 1
* From the equation
# a² = 16 ⇒ a = ± 4
# b² = 9 ⇒ b = ± 3
# h = -1
# k = -5
∵ The vertices are (h + a , k) , (h - a , k)
∴ The vertices are (-1 + 4 , -5) , (-1 - 4 , -5)
* The vertices are (3 , -5) , (-5 , -5)
∵ c² = a² + b²
∴ c² = 16 + 9 = 25
∴ c = ± 5
∵ The foci are (h ± c , k)
∴ The foci are (-1 + 5 , -5) , (-1 - 5 , -5)
* The foci are (4 , -5) , (-6 , -5)
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