What is the equation of the hyperbola vertices (-4,0), (4,0) and foci at (-5,0) (5,0)

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

* Lets solve the problem

- To find the equation of the hyperbola we need the values of a² and b²

∵ The coordinates of its vertices are (-4 , 0) and (4 , 0)

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

∴ a = 4 and a² = (4)² = 16

∵ The coordinates of its foci at (-5 , 0) and (5 , 0)

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

∴ c = 5 and c² = (5)² = 25

- To find b use the rule c² = a² + b²

∵ c² = a² + b²

∵ a² = 16 and c² = 25

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

∴ b² = 9

- Lets write the equation of the hyperbola

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

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