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
* 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
