Answer:
The equation is (y + 3)²/4 - (x + 5)²/36 = 1
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 center of the hyperbola is (-5 , -3)
∵ The coordinates of the its center is (h , k)
∴ h = -5 and k = -3
∵ Its vertices are (-5 , -5) and (-5 , -1)
∵ The coordinates of its vertices are (h , k + a) and (h , k - a)
∴ k + a = -5 and k - a = -1
∵ k = -3
∴ -3 + a = -5 and -3 - a = -1
∵ -3 + a = -5 ⇒ add 3 to both sides
∴ a = -2
∵ Its co-vertices are (-11 , -3) and (1 , -3)
∵ The coordinates of the co-vertices are (h + b , k) and (h - b , k)
∵ h = -5
∴ -5 + b = -11 and -5 - b = 1
∵ -5 + b = -11 ⇒ add 5 to both sides
∴ b = -6
∵ The equation of it is (y - k)²/a² - (x - h)²/b² = 1
∵ h = -5 , k = -3 , a = -2 , b = -6
∴ The equation is (y - -3)²/(-2)² - (x - -5)²/(-6)² = 1
∴ The equation is (y + 3)²/4 - (x + 5)²/36 = 1