Write the equation of a hyperbola with a center at (-5, - 3), vertices at (-5, - 5) and (-5, - 1) and co-vertices at (-11, - 3) and (1, - 3).

The equation is (y + 3) ²/4 - (x + 5) ²/36 = 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 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