Answer:
The equation of the hyperbola is (y + 5)²/16 - (x - 3)²/81 = 1
Step-by-step explanation:
* Lets revise the equation of the hyperbola
* The standard form of the equation of a hyperbola with
center (h , k) and transverse axis parallel to the y-axis is
(y - k)²/a² - (x - h)²/b² = 1
- The length of the transverse axis is 2a
- The coordinates of the vertices are (h , k ± a)
- The length of the conjugate axis is 2b
- The coordinates of the co-vertices are (h ± b , k)
- The distance between the foci is 2c, where c² = a² + b²
- The coordinates of the foci are (h , k ± c)
* Lets solve the problem
∵ The vertices of the hyperbola are (3 , -1) , (3 , 9)
∵ The coordinates of its vertices are (h , k + a) and (h , k - a)
∴ h = 3
∴ k + a = -1 and k - a = -9
∵ The co-vertices of it are (-6 , -5) and (12 , -5)
∵ The vertices of the co-vertices are (h + b , k) and (h - b , k)
∴ k = -5
∴ h + b = -6 and h - b = 12
∵ h = 3
∴ 3 + b = -6 ⇒ subtract 3 from both sides
∴ b = -9
∵ k + a = -1
∵ k = -5
∴ -5 + a = -1 ⇒ add 5 to both sides
∴ a = 4
∵ The equation of the hyperbola is (y - k)²/a² - (x - h)²/b² = 1
∵ a = 4 , b = -9 , h = 3 , k = - 5
∴ The equation of the hyperbola is (y - -5)²/(4)² - (x - 3)²/(-9)² = 1
∴ The equation of the hyperbola is (y + 5)²/16 - (x - 3)²/81 = 1
